Its club charter started out as a copy-and-paste from the 501st charter with the names replaced, but it has evolved since then.
The Rebel Legion is also recognized by Lucasfilm, but you don't hear that as often.
In both clubs, it is the agreement with LFL that requires that all and any proceeds go to charity.
There are some 1700 members worldwide. I get the impression that there are more women in the Rebel Legion than in the 501st, but don't quote me on that.
Membership is not exclusive - if you have the costumes, you can be in all clubs you want as long as you follow the rules.
If you have Clone armor, then you can also have it in the Rebel Legion - until you get order 66. ;) Even the "501st" clone armor is allowed!.
Anakin Skywalker Episode III costumes belong in both clubs, except that in the 501st you must have "Sith eyes" and in the RL you must not.
In some regions, there is very good cooperation between the local RL and 501st units at events. In others, cooperation is not so good.
However ... some garrisons of the 501st have also created a rival detachment, called the "Rebel Command". As far as I have heard, and seen of 501st members in Rebel outfits at events, these do not have as high costuming standards as the Rebel Legion.
The "Rebel Command" is disliked by many Rebel Legion members, as they are often confused, and are bad for the RL's reputation of being a club with only high-quality costumes.
There is also a costuming club called The Jedi Assembly for Jedi. I don't know much of them.