What happens when you vote up comments? Does it give points or something?
There is no reputation for comments.
However, when there are more than (usually) five comments, only the top five upvoted comments are displayed and the other comments are only loaded upon clicking on "show [X] more comments".
Upvotes help making people aware of "good" comments (where "good" usually is helpful, informative, etc.). Some people also upvote to indicate that they received/read a comment addressed to them (rather than adding another comment "@user Good idea, thanks.").
Adding to @Earthliŋ♦'s great answer, there is a silver badge called Pundit in Participation Badges that you can get when you have 10 comments with score of 5 or more.
It takes a long time to get this badge, but the more helpful comments you leave, the shorter it will take. In other words, the more upvotes you cast for other users' comments, the sooner they will get the badge.