There's no real difference. As Su.Ki. has pointed out, "bye-bye" is slightly infantile. When it's used, it's usually by women. "B'bye" is fairly common, too.
In the U.S. when people say "bye" the usually stretch out the word a bit with a little change of pitch "Byyyy-ayyye!"
The underlying meaning of "bye" is short for "Goodbye," which is a contraction of "God be with ye," which is short for "May God be with you until we meet again." Most parting expressions combine the idea that you will meet the other person again, and that they will have good luck until then.
In the United States, it is very common to say "So long." (Nobody seems to be completely sure, but this may be a variation of Arabic "Salaam" and Hebrew "Sholom.") "See you later" is very common, too.
What people choose is a combination of what's common in the region, what their family used, an expression of individuality, etc.