Mezzie and Georg are correct in that "bring it on" is usually used in competitions or fights. It can also be heard in the workplace. For instance, if there is new workload presented to you, you could say: "Bring it on" as accepting of the challenge (much like the above answers imply), or "Bring it on" sarcastically, as "Sure, dump even MORE work on me."

I've only heard it in the context of fighting and competitions, and it's always used to provoke an opponent. "Come on" can be used in that situation, too, but "come on" can be used in a variety of ways outside of that context as well, such as to mean "hurry up" or "let's go." "Bring it on" doesn't have that flexibility. If you use it, it's either to rile someone up or as a joke. Because of that, I can't give you any examples of other uses for that phrase. Who knows, though -- maybe in another English-speaking area it IS used in other ways. We'll see if someone else replies. :)