The song does differentiate between billiards and pool. If you watch the movie, Harold Hill asks what is going at the billiards' parlor and is informed the Iowans have a brand new pool table. He then goes on to sing that he considers the hours he's spent playing billiards to be productive, but that pool will ruin the town's youth. He then continues to speak of pool as a game with six pockets and 15 numbered balls.
Ya got trouble Yes, ya got lots and lots o' trouble I'm thinkin' of the kids in the knickerbockers, shirttails, young ones Peekin' in the pool hall window after school Ya got trouble, folks, right here in River City With a capital 'T' and that rhymes with 'P' And that stands for 'pool'!
More here. As you'll see, the song appears to use billiards and pool to mean much the same thing.