The "what's the difference" question has been argued every which way before. I think it's important, for the sake of discussion, to remember that as with any phrase in any language, "Old Time Music" has come to mean certain things in our language and most importantly, amongst the folks who actually use these words to define the music that they play and sing.
Most (but not all) OT played and recorded today is what would have been played for a dance. As Pete said, the "song" branch of OT has been neglected somewhat in favor of the "tune" branch. The same can be said for the picking styles in that there has been a certain narrowing down in that department too.
BG music features instrumental breaks by the musicians. OT is generally played ensemble style without these lead breaks.