You have a different colored Pokemon that is really rare. Bragging rights.
Shiny Pokemon do not have any special skills or stats. They are just really hard to find... A shiny Pokemon is like a trophy; while it is not any stronger than its normal version, the fact that you have one is pretty cool. The chances of finding a shiny Pokemon is normally 1 in 8192. That's a 0.0122% chance of finding one! Don't bother breeding for one either, as the chances of receiving a shiny egg are just as slim.