USAG is an internal American program. FIG is the code of requirements followed by gymnastics worldwide. American gymnasts competing elite and internationally compete under this code. I think everyone else in the world follows FIG (or modified versions of it) for most of their development system. In general penalties are harsher under FIG and requirements higher.
Some of the Y programs compete against other YMCA's and some against ordinary clubs, there is also a wide variety of programs that they compete in and also the standard of their programs. MaryA has a dd in YMCA and probably knows more about it.
I Have heard there are also some other great programs in the US like USAIGC (? is that right) JOGA , XCEL, and the Mason Dixon league for eg.