Mathematicians regard the Riemann hypothesis as very important, not least because a fair number of important theorems have been shown, given the assumption that the Riemann hypothesis is correct, so as soon as it's shown to be true, a whole pile of other interesting stuff is also true. Its impact is both practical and theoretical.
But when I was talking to William earlier today, he summed it up well:
"If it isn't true, math is a lot weirder than we think it is."
[Oh, and in case you are wondering what the situation with Li's proof is - while I was away on vacation for a few days, the paper was withdrawn. It appears Li could not rescue the proof. ]