Questions that invite mostly opinionated answers are pretty frowned upon on all stackexchange sites as you are aware, but I do so understand why sometimes you just can't help yourself sometimes. You need an answer to something you are thinking even if it's mostly philosophical, there's some chip on your shoulder and you need to put an end to it.
So I will try to actually answer this question for you in a way that I hope will satisfy you, put a bit of a rest to this debate, and most importantly, put your mind at ease. Sorry for the length of my answer, but questions like this require a bit philosophical and metaphorical answers to actually qualify as good answers I believe. (Vague and broad questions get more detailed and complex answers)
So, Traditional Martial Arts that were developed to face real life threats under real life circumstances with very real pressure, often life versus death. I like to think about traditional, Japanese Ju Jitsu as an example of this. Might just be that this is what I have practiced myself the most. This martial art was developed for the purpose of farmers being able to stand up for themselves when threatened by the often tyrannical samurai of their time period. (The modern day equivalent I believe would be Krav Maga)
And Traditional Martial Arts's beloved child, Competitive (Sport, Showmanship) Martial Arts. This will be Judo (I mean that is exactly what judo is, a competitive version of the traditional ju jitsu I spoke of before) and Pro Wrestling (Pure focus on showmanship, little to no focus on actual fighting, it's all staged, and it's all meant to be staged). I won't actually give pro wrestling any sort of focus since it can easily be argued whether or not it is a martial art to begin with (but wrestling is, and it is just as much sport martial art as judo based on mostly similar principles) I just felt like throwing it out there, that that kind of martial art exists, and that I think that that is ok even if I personally wouldn't participate in that.
Is one better than the other?
No. No they are both quite great in their own right. Think of it as a tool, when you want to hammer a nail into a block of wood, do you use a hammer, or do you use a crowbar? You obviously use the hammer designed for this task right? It's not like you can't use the crowbar to do the same task, it's just that the hammer fits the job better, and does it better, both tools are very similar in nature and can often be used interchangably if need rises despite their very different design goals. (I'm not using either tool as a metaphor for a specific genre of martial arts, you can think of the hammer as traditional and crowbar as competitive, or the other way around as you wish, the fact is that it doesn't actually matter, both can be used as blunt force to hammer in the nail, both have a "claw" that can be used to pull the nail back out, and both can be improvised to force things open in creative ways. There always was the option of both in one, with a few extras which I shall call the haxbar, but that's off topic)
So let me turn the question around to you. Why are you practicing martial arts?
You want your body to be in good shape? Well great, martial arts are great for that and you know what, it doesn't matter at all which martial art you are practicing if this is your goal. You may however want to focus on striking based martial arts (rather than something like Judo or the many other variants of Ju Jitsu) because the striking based martial arts tend to have a heavier focus on physical fitness than martial arts that are more technical.
You want to be able to effectively fight anybody 1v1 (unarmed vs unarmed; relatively fair fights)? Well, now we're entering competitive martial arts's strongest area, so you know what, if this is your goal, then you really shouldn't worry about whether your martial art is sport or martial art, both will work very well, but often competitive martial arts are more tailored towards this kind of fight and thus I think that would be the logical choice here. However, just like with the crowbar and hammer, the traditional martial arts will tend to work very well in these situations too, but don't be surprised if you lose to a competitive martial artist (one of the primary reasons you would is not because the martial art is necessarily better, but people who are practicing competitive martial arts will often spar more, and spar more seriously, as in actually enter a ring and fight according to a set of rules, they have more experience in this than you). But I can easily turn this around and say that the rules of the competitive sport might hold back the competitive martial artist who would lose to the traditional martial artists when the latter decides not to play fair (A popular example of this would be that boxers never kick, this is because the rules of box dictate this, and thus they become limited in fights where there are fewer rules) but more often than not I believe the competitive martial art will really come through here, they are tailored towards these kinds of scenarios, traditional martial arts will however often work just as well because since they are thought through without any rules, they may have a more open mind to stepping outside the conventional ways of fighting that the competitive fighter may end up being mentally locked into. It's all very circumstantial.
You want to be able to more effectively fend for yourself in tough situations that arises in life (like assaults and street fights)? Turning around the previous example, what if the fight is not fair? While this is great too, and you will get better in this scenario from almost any martial art, again because the fact is that any solid fighting style (competitively based or street fighting/traditionally based) is going to help you in those situations. Some may help you more than others thought so this is where the question becomes important. For example, we are no longer dealing with unarmed vs unarmed, you may end up in a situation where you unarmed have to fight against someone who is armed, and you need not only to have the techniques to deal with that, but also the mental preparation (don't fear the knife, for example). You may also be in a situation where you are 1 and they are 3, many martial arts cover these kinds of scenarios, but they are usually more heavily focused on in traditional martial arts. For example in Japanese Ju Jitsu trainings for these situations may come very early (e.g. for total newbies, yellow belts and the like). So you will also be more mentally prepared for this with traditional martial arts than competitive. Remember that having the right mentality and knowing the right responses to deal with any combat situation is half the fight to clawing your way out of even the most grim of situations. Don't get me wrong, I'm not saying competitive fighters are weak minded, but they haven't been trained for scenarios where their life is in danger. It wasn't important in a ring to be prepared to take on a knife, or even a gun (Krav Maga does that...). Not all competitive martial arts are created equal however, and I think for example Kickboxing, Muay Thai or Taekwondo might save your neck in many (or all) of these situations, but Brazilian Jiu Jitsu would be like an Achilles heel in a street fight where you have to fight against more than one opponent. (It will be pretty easy while you are restraining one person for the other to kick you in the face). It's again all very circumstantial, but there is another thing many traditional martial arts (including again the famed Japanese Ju Jitsu) and that is training with a wide variety of weapons, so if you have this kind of stuff under your belt, you may get attacked by an armed person, you get to disarm him and use his weapon against him (and his friends), but for a competitive martial artist who hasn't trained to do that his fists and legs will always be a much more effective weapon than any object he will be able to rip from the hands of a knocked out or incapacitated opponent, and this lack of versatility can really hurt your odds sometimes.
Do you want to fight professionally? Do you love fighting (or martial arts) so much that you want to do it for a living? Is there some thrill you get out of fighting in front of an audience? Proving your mettle to thousands or even millions of people by beating a strong opponent? Or narrowly losing to a strong opponent after a drawn out fight? Do you like the romance and drama surrounding ring fights? The connection you can only make with another person by fighting them? Etc. Do you want to fight as much and as often as possible? Any or all of the former? There are two valid routes here. The most obvious and highly recommended one is going all out in competitive martial arts. In a ring fight, traditional martial arts may end up holding you back because they might utilize a style that bends or even breaks the rules of a given ring, for example Japanese Ju Jitsu while just as Brazilian Ju Jitsu it focuses a lot on locking down and restraining/choking an attacker until he submits, however Ju Jitsu also has a plethora of moves that can permanently incapacitate, maim or even kill people, and this is not something you are allowed to do in a ring fight, it is also something you would extremely regret doing in a ring fight. You may have lost your temper and just done it by accident, but you don't have any undo button so it's better for you not to know how to do that, not to have an instinct to do things like that in a ring fight at all rather than to know how but try to avoid it, reduces the risk. You might wonder why some self-defense focused art like Ju Jitsu would have moves like this, and it's quite simply because in the real world, outside of a ring, you will more often than not if assaulted, be assaulted by more than one person at a time and under circumstances like these you need to be able to think fast, act fast and if you can't run you need to be able to take all of them down fast (basically unfair fights as I talked about above). Brazilian Jiu Jitsu style joint locks and choke holds which are extremely good in a ring, would not do you very much good in a 1v2 or 1v3 scenario as I talked about before, it would be absolutely meaningless in a 1v6 or 1v7 scenario as well (on the other hand, things like muay thai, kickbox and taekwondo, striking arts, these will serve you pretty well in all these situations despite being competitive martial arts. Muay Thai was also notably not originally a competitive martial art, and it was very lethal, and even if you learn competitive muay thai, it only takes a minor twist to turn it from "sport" to "lethal". The same does apply to taekwondo, but over time it has become more convoluted by shifting it's focus to not only competitive but also showmanship. You know all the flying kicks, but as I noted in an answer I wrote yesterday to another question, even that may serve you well in a street fight because it can double as an intimidation technique. Will be no use against trained opponents though.)
Do you want to be the other kind of professional?
Going on from the above, there is another frequently overlooked use for martial arts here, Traditional martial arts do have a use in some serious jobs (not sports) that may involve you getting into a lot of fights, but these aren't as light jobs as sports, competing in a ring, these are jobs that involve real world combat scenarios and they range from anything between being a security guard or bouncer to being in a military or a mercenary, and in here you will want traditional martial arts having your back rather than sports because the situations you will end up in will usually not be as idealistic as "1v1 unarmed duels" which practically all competitive martial arts (except a few niche ones like kendo, which focuses on sword vs sword) are focused on. Don't get me wrong here, having a competitive martial art will do you a world of good (maybe save your life a couple of times) in this kind of a job as I said before, having any sort of martial art under your belt will do you good in any combat situation, ring or no ring. But when you're fighting in unpredictable environments against unpredictable opponents and unpredictable numbers of opponents which will unpredictably and quite variably be either armed or unarmed (knives, guns, pipes, baseball bats...) which you aren't going to see in fair competitions, do you want to have brazilian jiu jitsu, or judo, or do you want to have traditional ju jitsu (which by the way also has armed combat exercises at higher levels; e.g. you could pick up one of these weapons that others brought to the scene and use it to increase your odds, something you won't learn in any competitive sport I know of). When you are potentially betting your life, the gravity of the question asked becomes much more severe, and important, and you need to be able to make an informed but personal decision on this matter.
Bottom line
Competitive and Traditional martial arts while usually nicely interchangeable have a different focus, and because of that may end up serving you differently under different circumstances. Neither is better than the other, and the most important part is what you are going to be using martial arts for, why are you practicing this stuff? If you know the why, you know what martial art or at least what class of martial arts would be the right one for you. (Btw something I didn't cover before, if it's showmanship, I totally think various styles of Kung Fu are the way to go! Much cooler than Pro Wrestling, but that's just my opinion)
Every last one of us needs to decide for himself what martial art he wants to practice, and why he wants to practice it. What are you going to use martial arts for?
There's absolutely no use in judging others for what style or genre of martial art they chose, they did it for themselves under their conditions, you don't have their conditions, you have yours.
But do decide for yourself what you want to do and act on it, because having questions like these on your mind and even leaning towards one answer but not acting on this feeling is going to be like having a ball and chain on your foot. You won't be able to fight as well within your conditions, within your martial art because you are too busy thinking about whether it's the right martial art for you or not. Whether you should continue, or go try to find another martial art.
Remember that your mind is just as much a part of martial arts as your body, when one of them is not balanced or having doubts, your performance starts to drop. (Look to Connor McGregor for example, here's a good video if you're interested in the mental aspect of his fights, how his mind helps him win, and how he was losing fights before he re-centered his mindset; take things with as much grain of salt as you will, but at least acknowledge that your mind and what, and how you think is of importance and it will affect you, potentially greatly. And accordingly, make sure your mind is at peace with what you're doing, not just in martial arts but in life in general. If you are having doubts about your life choices, your life, or just yourself, it's going to bite you in the ass, repeatedly, until you do something about it.)
When you know what you want, look for the right martial art for you. For me personally, I believe that Krav Maga and Japanese Ju Jitsu for defensive techniques (and the ability to snake my way out of the most extreme situations, including being held at gunpoint) coupled with Muay Thai for a good and efficient striking capability would be best for me. I sadly can't practice Krav Maga, and I dropped out of martial arts a few years ago because of injury. But if I will go back into martial arts this is where my focus will be even if I'm actually pretty open to doing competitive arts like boxing (which I may very well do to get my body back into shape; it's more accessible to me than any of the others because there's a boxing gym at jogging distance. And I am ok with doing that for a while until I have the access I want to the martial arts I desire, I might even start competitively boxing, but it won't ever be more than a hobby to me, I don't want to be a professional martial artist, it's just not my thing, but I don't judge anyone for wanting that, and I don't doubt their choice of martial arts because I acknowledge that they probably just chose whatever they thought was best. And I am free to disagree with that choice as he is free to disagree with my choices, but we shouldn't have to argue about something like this; and in fact arguing over something like this is immature.)
Make your own decision, by yourself, for yourself, stick with it, and do not reject the fact that many others are practicing martial arts for different reasons than you or simply have a different perspective than you. As can be deduced from my long wall of text, both traditional and competitive martial arts are useful in all combat situations and it is very easy to argue which is better without finally reaching a conclusion, because there's always one niche or one corner case where one will be better than the other in whatever situation you can think of.
There are also some things to be considered further (your homework) that I didn't cover in much detail (for example competitive martial arts tend to have a lot more sparring, and competing in a ring gives you real fighting experience to which the only alternative for traditional martial arts is fighting in the streets in potentially life threatening situations; I mean if you can safely get through the process of amassing experience through the latter means then you sure will be a force to be reckoned with. I mean think about it, how much sparring can you really do with techniques that might with a slip of your wrist maim your sparring partner? When the martial art isn't centered towards a ring fight, sparring becomes harder to put into practice because of it's own nature. This is very inconvenient for traditional martial artists for example. And there are more hurdles like this for both ends as well as less visible benefits of both. For example the fame being a competitive sportsman can bring you, and attention it can draw to martial arts, increasing public interest; and the opposite, for traditional martial arts may also be a benefit to some (be that ninja, you secretly have a black belt in some obscure martial art and none of your friends know it, then a tough situation arises and nobody knows what the hell you did to overturn it; and you don't want them to know, since it gives you an edge to be an unknown)
Martial Arts are "Arts" for a reason (it's not "How to fight" it's "Fighting Creatively", this is also why at it's core martial arts are not a sport like football and probably one of the major reasons questions like this crop up every now and then. Never forget that, you're an artist, combat is your canvas, not your struggle the struggle is getting good at it; and just like with any other form of art, there are many styles, none particularly better than the other usually, just made by people with different tastes and perspectives of life).
But most importantly, don't forget. You're supposed to have fun in life! ;)
