What do I get from a skill being Legendary?
There are a few perks to making a skill legendary:
Since your perk points reset within that skill tree, you can reallocate those points within the tree or place them into another tree
Resetting the skill back to 15 allows you to continue levelling using a tree that you actually use in the style you play the game with instead of forcing you to max a tree that is not within your playstyle. For example, I prefer to play a stealth/ranged damage character, so don't particularly care for tanking so I could use the legendary skill feature to remove the need to max the block and heavy armor trees.
As you stated, the legendary skill feature effectively removes the level 81 level cap, so that you can attain ridiculous levels. A side effect of the level cap removal is that you continue to gain a perk point each level. You could theoretically get to a high enough level to have enough perk points to get every perk from every tree.
Source: http://elderscrolls.wikia.com/wiki/Skills_(Skyrim)
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.