To rebut is to try to prove something isn't true, but to refute is to actually prove it isn't. Getting them mixed up won't get you kicked out of the debate club, but it's worth knowing the difference.

‎To rebut is more than just "to deny," it's a serious attempt to prove something is false. If you rebut something, you haven't necessarily won the argument; you've only presented your side of it:

A Jesuit priest has taken to YouTube and his blog to rebut arguments raised by British physicist Stephen Hawking. (Catholic News Service)

Baldwin has used his Twitter account to rebut criticism. (Washington Post)

Refute means to disprove something straight up. If you refute something, you successfully win the argument:

The hearing was unfortunately based on hearsay evidence that we were not able to refute by cross-examining anyone. (Eagle Tribune)

Daly refuted the report Monday, saying that only one general manager had expressed concern. (New York Times)

So, if you try to defeat an accusation, you rebut it. You still need to butt in again, so you re-but, as in "But wait! I can still prove it!" If you actually defeat the accusation, you have refuted it.