As a graduate of one of them, I'd like to think that there is a bit of self-selection by personality / philosophy into each school. If I meet someone in a meeting that went to one of those three, I can usually determine which one they attended based upon their general demeanor.

That said, I wouldn't say there are large philosophical differences between them.

How would you describe the people who go to each of those schools?

Well, this would be playing a bit on stereotypes. But HBS students tend to be very management focused, Wharton students more quantitative / financial, and Stanford students more tech focused / creative (for an MBA). There are obviously tons of counter examples to each of those stereotypes, but they've held largely true in my ~9 years of experience in those circles.