Richard Lipton recently [blogged](http://rjlipton.wordpress.com/2010/08/18/proofs-proofs-who-needs-proofs/) about this question in the context of why a potential proof of $P \neq NP$ would be important.  I am probably bastardizing his words, but one of the reasons he gives is that a proof may give new insight and methods of attack to other problems.  He cites Wiles' proof of Fermat's Last Theorem as an example of this phenomenom.