A Billy Faggot is someone that acts like a faggot with the intent to piss someone off. A regular faggot is someone that just naturally pisses people off because that's their demeanor.
Steve: Trey was being a such a faggot last night.
Jill: No, he was being a billy faggot.
Steve: A what?
Jill: Trey was clearly acting like that to piss us off.
Steve: Yea, true. He usually doesn't act like Steve-o from jackass.
Jill: Exactly, Steve-o is just a regular faggot.
Steve: Right, because Steve-o is like that all the time, and Trey is only like that when he wants to be annoying.