Hello PH, welcome to Using English!
First, the literal meanings:
1. You "wind up" a mechanical device. This may involve a key (as in a clockwork toy) or a spindle (as in a watch).
2. Another meaning of "wind up" involves string, thread, twine, etc. For instance, you may "wind up" loose string into a ball.
3. You "wrap up" a gift, or an item to put in the post, or a purchase in a shop. Usually, you "wrap up" something in paper of some kind.
4. You also "wrap up" in cold weather. This means to wear extra clothing, to keep warm.
Now the metaphorical meanings:
5. If you "wind up" someone, it means you're teasing them, in order to make them annoyed. This meaning is a metaphor based on #1 above.
6. If you "wind up" a meeting, etc., you bring everything to a conclusion. This meaning is a metaphor based on #2 above.
7. If you "wrap up" a meeting, etc., you also bring it to a conclusion. This meaning is a metaphor based on #3 above.
The difference between "wind up" in #6 and "wrap up" in #7 is one of implication. If you "wind up" a meeting, the usual implication is that you have gathered together all the loose threads, i.e. finished the necessary business.
If you "wrap up" a meeting, however, there is no such implication: the meeting has simply been brought to a close – perhaps abruptly, or before all the business has been finished.
All the best,
Student or Learner