"Show tunes" are songs composed specifically for works of musical theater. A "soundtrack" is a commercial release of songs that accompany a film. The songs on a soundtrack may or may not be original. Many films use old songs. But if there are multiple original songs composed for a film, that music is collectively called "the score."

I agree with Jonathan's answer here.

In my opinion show tunes sounds a bit old-fashioned, but I don't think there is a big difference to the average English speaker. Perhaps for someone who produces music professionally there might be some difference.