Good bye is the most formal. Bye, is the least formal, bye bye/bye-bye are like the same thing...I wouldn't say there's any difference. Good bye is probably the least flirtatious... but hey, what do I know? I'm just guessing.
I actually want to know what girls think when guys use them. Like what would you think each one means? For instance, I don't like my dad and before I hang up I just say "Bye" or nothing. But if I'm talking to my friend that I really like I'll say "Bye bye" or "Sleep well babe." What do you girls think of different good-byes?