As a totally unbiased poster with no conflict at interests as all (I'm certainly not a Columbia student or anything like that), I'd definitely say that Columbia is the best school 😀.
I'll try to talk about the schools in a columbiacentric manner, but unfortunately I only applied and interviewed at Cornell, Columbia, and NYU so I can't speak directly about Mount Sinai or the others. I have heard lots of great things about Mount Sinai though, and my interactions with the students there have been positive. Lot's of cross talk between them and Columbia, more so than even with Cornell (which is maybe counter-intuitive since there are a lot of academic and professional connections between our schools).
Location: Cornell, NYU, and Mount Sinai all have great locations, but they also have crappy logistics in terms of the subway routes (I'm not entirely sure if this is true for Sinai, but the other two aren't even near a stop. That's a HUUUUGE hassle in the life of a medical student and an interviewee if you aren't taking a cab). Columbia is attached to the A line on an express stop, so even though we are a little farther away from the action we tend to have about the same transit time with less walking involved when going to places of interest in Manhattan. This is somewhat less true late at night, when the A runs local and adds another 15+ minutes to your traveling time.
Prestige (not reputation): Well, I don't think anyone can truly argue that Columbia and Cornell are the most prestigious of the medical schools, being Ivy League schools and also top 20 USNWR schools, the two things most likely to be known by the layman. NYU and Mt Sinai are well respected here in NYC though, so the day to day wow factor will still be there if that is what you crave.
Academic/Profesisonal Reputation: Columbia-Cornell (NYP hospital system) are widely respected as top research and clinical schools (not USNWR rankings). Part of this is because, among the two of them, they gobble up many of the best hospital affiliates in the city, in terms of research impact, community outreach (Harlem Hospital and also a lot of world famous community clinics like the Young Men's Clinic), and just sheer magnitude of clinical services. Bellevue is famous, yes, but I'm not sure how regarded it is by the medical community, with the notable exception of their ER. Mt. Sinai has a great hospital, and is like NYP in its ability to provide health care to both the richest and poorest of New Yorkers. I don't know much about the others, but Albert Einstein is affiliated with Calvary Hospital, which has some unbelievable stories to tell. Other medical schools do send kids over for clerkships and such though, but I feel like Einstein gets a lot more of it.
Public Health: Columbia owns this as far as I'm concerned, especially if you are interested in getting an MPH. Bellevue gets points too. Cornell and Columbia have fabulous international programs and opportunities, many of which you can do regardless of which school you are in. I can't speak for the other schools, but Columbia has amazing affiliations with programs like Harlem Hospital, the Young Men's Clinic, the Door, etc that you can do your first year clerkships in (shadowing plus a little extra basically).
Cost: Definitely not Columbia/Cornell, though there are scholarships. Interestingly enough, older non-trads get amazing financial aid from Columbia, and the need based isn't bad for the rest of us either. It's just that you have to have quite a bit of need to fill out that 30K+ a year tab. However, in general NYC isn't the cheapest place to begin with unless you are living at home.
Student body: Most of the students I've met from other schools have been great. I haven't met a lot of Cornell kids, but that might because there are less of them or because of their grading system.
I'd also consider things like P/F and ranking of your pre-clinical years. Also, just the format (Columbia's is 1.5 instead of 2, etc). Research is great and all, but quite honestly unless you are getting a PhD I wouldn't worry too much about it. ALL of the schools here have butt loads of research. It really doesn't matter that much if they have 100 million or 200 million dollars in research funding, because you are only going to be getting like 10K of that if you are lucky in the end anyways.