I'm an incoming HPSP M1, not a doctor. The following is secondhand, and is probably at least partially wrong, so do more research.
Army: Straight through training, very few PG1s get sent to practice before competing residency. Almost no chance for a civilian residency, because they need to fill their own very extensive residency system. Currently 15 month deployments.
Navy: Very high chance of a GMO tour. The recruiters say they will go away before it matters for you, those with experience on this board seem to suspect that that either won't happen or will only get half way done. They won't let you take more than one ADT per fiscal year, which is a shame (I think) because you´ll probably need to use one on school orders. Apparently they have some really extensive pilot training, compared to AF or Army, if you decide to go the Flight surgeon route. More flying time.
AF: have heard no positive reviews, though again I've go no personal experience. Apparently there are accredidation problems with a few of their residencies? The number of residencies also apparently can vary wildly from year to year in some fields (because they rely on civilian residencies more than other branches, they can 'cut' the number of civilian deferments allowed in different residencies from year to year, so one year there are 30 EM residencies, others there are 3 and its suddenly as competitive as neurosurgury). And while they have fewer GMOs than the Navy, they're not even pretending to want to get rid of them. And finally, from the experiences of others on this board, this is the only branch that might not give you the schoarship at all (assuming you want to go HPSP, read up on FAP before committing).
My Opinion: on paper I think Army looks best. I've wanted to be Navy for long enough that I'm going Navy anyway (I like ships and want to work with the Marines), but if I was choosing for purely practical reasons I think I'd go Army.