If they REALLY want to compete with each other, I believe the best way to snag applicants would be to "out-scholarship" each other. That could definitely win me over! Hah!
Anyhow, if your stats are good, both schools will want you (and they've clearly demonstrated interest by offering you interviews). I don't think that interviewing at one would make the other one drop you. If anything, the fact that you're interviewing at multiple schools means that you're a desirable candidate. Why would they want to risk losing you just because you're interviewing elsewhere? It seems like they would rather end up with those candidates who had multiple options because they were so great.
Call the schools, and it's probably a good idea to be vague, but don't lie (just like LBJ6 said!). I think this is a no-brainer, too, but make sure you first try to reschedule with the school that is your second choice. Best of luck!