@DreamsUnlimited, Of course there are other great schools, but to include all of them would be crazy! There would be a huge list to sort through, so I just limited it to Ivies.
Dartmouth is known to be preppy? I've never heard that before, but that's just me. Columbia is full of hipsters? I know Brown has lots of "hippies" and hipsters, but didn't know Columbia was the same. I agree with you, Lobbz, that HYP are the most well-known. They're the magical triad.
The only two that instantly strike me as groupable together are Harvard and Yale. Other than individual departments' strengths and weaknesses, it is very difficult to find meaningful differences between Harvard and Yale colleges as a whole. They are distinguished from Princeton because of their campus types and Princeton's greater focus on undergrads.
Brown is certainly unique because of its open curriculum. Dartmouth is unique because of how in the middle of nowhere it is (like Princeton and Cornell) but it is distinguished by its social (see: drinking) scene being better than the other Ivies' (rough generalization but mostly true).
There's a huge difference between Princeton's suburban location (less than one hour to Manhattan or Center City Philadelphia on public transportation, lots of amenities nearby, within an hour's drive of maybe 12 million people), and Dartmouth's or Cornell's much more isolated location. And even between those two -- Dartmouth is in a tiny town, but you can drive to Boston in a couple hours and a bit; the only cities you can get to in a couple of hours from Cornell are Syracuse or Binghamton. But Ithaca itself is a much more substantial place than Hanover. It has something like 30,000 college/grad students during the school year (over 4x what you would find in Hanover), lots of restaurants, a gorgeous lake, wineries, etc.