This is sort of a lost cause, as I doubt Sabyaschi will want to come back given his cold welcome, but let's take a look. I'm going to ignore the latest iteration and actually go with the original question.
Given the assumption that there are multiple TKD and multiple Karate schools available, with differences among each, here's what you're looking at for differences:
From a distance, if you squint, you won't be able to tell TKD apart from Karate, unless you see someone wearing shoes, in which case it's definitely TKD.
There are two major TKD federations - ITF and WTF. WTF is used in the Olympics, and is what most people think of when they hear or say TKD. The rules of WTF TKD prohibit punches to the head, only allowing punches to the body, but allow kicks to the head and body. Punches are also not scored as highly as kicks, so as a result of the rules, WTF TKD is known for its kicks. It is full contact, and the dipped foam headgear they use is worse than useless, so there is a high risk of concussion.
ITF is 'semi-contact' which is rather hard to define, but in general, you're not supposed to knock anyone out. Since it's semi-contact, the reasoning for disallowing head punches in WTF (risk of opening a cut) doesn't apply, and ITF TKD is much more balanced with kicks and punches than WTF TKD. In fact, I'd go as far as saying ITF TKD is more alike with JKA Shotokan than it is different.
There are other TKD styles, but ITF and WTF are so dominant that I've rarely come across a school that isn't one of those two.
With Karate there's more variation, but loosely they could be broken up into Knockdown, No/Light Contact and Non-Competetive.
The most prevalent example of Knockdown Karate is Kyokushin. It features no protective gear (except for cups), allows kicks, knees, punches and elbows to the body. Only kicks to the head are allowed - knees, elbows and punches have a high risk of opening cuts. Fighting is largely a matter of attrition. You'll either drop out or develop a body that's very tough in being able to take hits. You'll also end up with bruises on a regular basis. There's a risk of concussions from being KOed with a headkick, like WTF TKD, but I'd say it's actually slightly less. Because there's no body armour (WTF body armour is actually pretty good, unlike the headgear), there's also a decent chance of being dropped by strikes to the liver or xiphoid process. Leg kicks are also allowed, and are a popular way of taking someone out of the fight. You'll certainly be at risk of being hurt, but the chance of concussion goes down.
For No/Light Contact Karate styles, you'll see it being mostly like ITF TKD. Because you're not attacking with real force, punches to the head are allowed. There's relatively little risk of concussion, or any serious damage in one of these styles, unless they're fond of punching Makiwara to toughen the fist.
Non Competetive Karate styles might have some form of sparring in class, but the emphasis will more likely be on forms, and despite the name, you might also practice with some weapons. You're not too likely to get hurt or acquire a concussion, but at the same time you're also not all that likely to learn how to fight.
If you're interested in learning how to fight well, Kyokushin is your best bet - just make a habit of protecting your head in competition, and also decline any invitations to compete until you feel you're really ready.
If you're interested in the cultural aspects of martial arts, a non-competetive style is probably your best bet. Either TKD or Karate would do, but again, I haven't seen any non-competitive TKD. If on the odd chance that there's a Taekkyon school where you are. Check it out, it's quite different from either TKD or Karate, and I'd say it has more in common with Capoeira than TKD, protestations of some TKD exponents to the contrary.