This is a bit nuanced, but the feeling of the ideas is different. If one were to say "you're not invulnerable", it seems the speaker is wanting to convey to the listener that he isn't invincible. However, if someone were to say, "you're vulnerable", it implies that someone is in danger right now. While the definitions of each word in the phrases could lead you to the conclusion that the phrases are saying the same thing, the circumstances in which you would use each phrase are different. Hope that helps a bit!

You would say it in order to contradict the person. You believe the other person imagines himself to be invulnerable. You want to contradict him - to scold him - by telling him that he is NOT. By using the word "not" you are going against him. It's like saying NO! If you say "you're vulnerable" there is no logical difference, but logic is not the point. It's about saying NO!

For emphasis.

There are many other examples where phrases that seems to have the same logical meaning, nevertheless have big differences in emphasis. For example: a) Not all expensive food is good. b) Not all good food is expensive. Logically these are exactly the same, but they feel so different you have to think about it. Sentence a) makes you think about expensive food that is bad, and sentence b) makes you think about good food that is inexpensive.

They are logically the same. It's a question of context. We often use the word "invulnerable" to describe a superhero like Superman. Suppose person X seems to think they are Superman. The direct way to oppose this is to say "You think you're Superman? No, you're not Superman." Saying "you're not invulnerable" is similar. "Invulnerable" is the starting point. We are thinking of the word "invulnerable," so we say "you're not invulnerable."