In short, not a whole lot. The only reason is to allow your character level to continue to rise (again, no benefit for this either).
Taken from the wiki
Legendary skills level up normally, and have no added benefits compared with non-Legendary skills. The purpose of making a skill Legendary is to allow it to contribute to increasing character level again. In Skyrim, character levels are gained by leveling up one's Skills. When all skills reach 100, the character can no longer gain character levels (this occurs at level 81). Making skills Legendary allows skill leveling to continue, which in turn allows character level to continue to rise.