I have an audio file (Music + Vocals) and I have an instrumental (Music Only). Is there a program or some way I can take the music file and subtract the instrumental leaving only the vocals? (Full Song - Music = Vocals Only)
Audio waves are additive, so if you play back the inverse of a waveform against the original waveform, they'll cancel out to nothing.
So if you have the exact instrumental, you can invert the waveform and play it against the original recording, which will mostly leave only the vocals intact. It's not the same as having the original tracks, since there won't be the same fidelity (you're subtracting from the mastered final so there's going to be a bit of loss of resolution) but this should get you pretty close.
You can use a waveform editor like Audacity to do this. You might have to shift one of your recordings forwards or backwards in time a bit to get the samples to line up; this is particularly the case if they came from separate sources that may have been encoded separately.
Happy remixing ;)
Although this is already answered I made a tutorial on this a while back also showing how you can do this in Ableton Live.
Here's a picture illustrating what Warrior Bob said:
There's a VST plugin to do basically this, called Extra Boy. I haven't tried it myself, so can't vouch for its effectiveness.
I find the hardest part of this is to actually fit the song exactly against each other. In a case where the song is not exactly the same for example an 'instrumental' version of the same song in the same album, it's already hard. Also in cases where you want to subtract the music from a video, it's practically impossible to line up the music. A video might even cause the music to be clocked slightly differently after being edited/converted. The only time it worked perfectly for me was when I used the same music compared to its inverted version.