you get a percentage of their land and population counting toward domination victory. or, if you don't have enough land/pop for that type of win, if you make everyone else your vassal, you get a conquest victory, so it's one step closer to that.
but before victory, you get the benefit of using their land for movement and healing, and it can help your research because you can tell them to research one tech while you research others. and if you are on good terms they will probably vote for you in any voting scenarios since they are now a team member. and you can usually ask resources from them for free or least get good trades. also you get extra happiness in your cities for having an influence on other civilizations (check in your city screen and mouse over the happiness faces at the top, there is probably a +1 happy for this reason).
really the benefits are pretty extensive - it's like having a team member but you do pay some extra maintenance and there can be other drawbacks depending on the situation of your game.