You're talking about financial aid dude--not potential buyers for a house. I highly doubt you can "play schools against each other" so that one school will match another's offer. Just take what they give and pick your best option.
You want to strike a tone that doesn't seem manipulative, but comes off as sympathetic and reasonable. It depends to some extent whether you're talking about merit or need based aid. If the aid is need-based, you might try to talk to the financial aid office with something like, "I got an offer of x from this other school, and I was wondering why your office generated so different an offer." You might be able to get them to revise their offer. If you got a merit scholarship, I'm not sure how much luck you'll have. It kind of depends on how much cajones you have. You could just say outright to the admissions office that you got $x from this other school and only $1/2x from their school, and that because of the amount of debt involved, even though you want to attend their school, you probably can't due to the financial imbalance. Since they've already admitted you, and the admissions office will have no real role in your med school career, there isn't any real risk to this. However, unless they want you badly, they probably won't up your offer much.