To hear the MGS theme in the first MGS, you're supposed to play through it and get one ending, save, replay and get the other ending. The music is supposed to change in the credits I believe. I don't know if this is retained in The Twin Snakes version though.
I thought someone would have answered this question by now.
Edit: The music change should play Metal Gear Solid Main Theme (1997 Edit) from the OST, which is track 20 on the CD, I believe.