The answer to this question entirely depends on the form and the history behind it.
Most forms have changed in small and often big ways since they were first created. Sometimes movements are repeated slowly just because someone thought it would look better in a demonstration. That's the truth for most of what you'll see out there when you see something being done slowly.
Dynamic tension is an exception. There's an actual purpose behind it. It's done slowly not for the sake of looking slow, but because you're tensing up your muscles and feeling the burn, basically. You have to go slow for that. How slow? It's timed with breathing, so that you perform the movement in one breath. And your breathing is also slowed, due to abdominal muscle contraction. I won't explain dynamic tension, but suffice it to say, there's a reason for it and why it's done slowly.
In the TKD video you linked to in your question, it shows a form where a kick is done slowly. This is done in TKD as a test of balance, flexibility, and your ability to remain calm. You generally don't see this in older martial arts, because that wasn't something they cared about as far as forms are concerned. They might have developed those attributes in other ways, just not in forms, because forms had another purpose.
Other times in forms you might see slowness done with little or no tension but with a coordination of breathing. That could be added after the fact (after a form was created). And it's usually added by someone who wants to incorporate a form of chi-kung / ki-ko to their martial arts. They're attempting to circulate chi/ki to their limbs. You'll see that a lot in Chinese kung-fu forms.
Slowness that's not due to dynamic tension, chi-kung, or as a test of balance, flexibility, and mental state, is there for emphasis only (or to look cool). Why the emphasis is there is generally lost to time. Who knows why the original maker of the form wants it there. People can guess, but without something in writing or without something passed on in oral history, nobody can know definitively the reason why.
To give you an example of why there's emphasis placed on a movement, it's sometimes done to mark the beginning of a throw. In these cases, this is the point in the form where you're emphasizing that the other person is now in trouble and about to be thrown. It's like saying, "Here we go!". You can see such an example in my breakdown of the Heian-Sandan form, here:
Name and meaning of stance where you stand with fists on hips?
In Heian-Sandan, you often pause a little after turning around and standing up in the middle of this form. The interpretation is that this is right after you've grabbed a hold of your opponent in preparation for an "o-goshi nage" throw. The pause is emphasizing that this is where the throw happens. It's a way of marking the end of something and the beginning of the next thing. In real life, you wouldn't actually pause there. You'd just complete the throw. And in the form, the pause generally doesn't need to be there. It's just that someone thought it should be there for emphasis. That's all.
Here's a video showing a demonstration of Heian-Sandan. The movement in question happens around 0:33...
My recommendation whenever someone asks why something is done slowly is to look back on old black-and-white movies or videos showing someone performing the form, and compare it against what the form looks like today. Also, compare different lineages and branches that all practice the same form and see how they differ. That can often lead you to discovering something. And of course, try to get good bunkai for it (look up "bunkai" if you don't know what it is).
Hope that helps.