You are asking a question about subsets and categories really. All ballroom can be danced socially but not all social dances are ballroom.
Social dance is something that people come together to dance, it can be in pairs, alone, formation, whatever, and purely for social reasons. Much of the ballroom you see on television is not social, but rather competitive or for artistic impression. Of course I've seen social dancing in the media, but it often falls in other categories.
Some may suggest that the beginning steps of each dance are the only social, but really they are the basic foundation of social so that the dancer can move down the floor. Many social dancers spend a lot of years learning amazing moves that are very advanced. I see the difference as being very much the motivation that drives the dancer.