Actually, being a current undergrad at Harvard as well as one of the Thiel 20u20 finalists, I have to pitch in my two cents and say that in immense part, what makes Harvard Harvard
is the fact that not everyone can go there.
The individual validation that a student gets upon attending Harvard is predicated on the unparalleled difficulty of getting in. In my class (2014), for example, admissions were at an all time low - ~7% admission rate, with ~30,000 applicants. Possibly a majority of those applicants could have attended Harvard and graduated with respectable success, but had Harvard chosen to open its doors to all of those applicants (just assuming for the moment that its resources would have allowed it to do that), the respect a Harvard admission garners would plummet - because it proves very little.
It no longer demonstrates that you're any better than your peers, and there's the problem. I'll shoot up a counterexample - when I was choosing between schools, Harvard was actually far from my first choice. UChicago was hands down the college I wanted to go to, and I had my heart set on it. In all honesty, the academic rigor at UChicago is lightyears above that at Harvard (saying that after now having taken courses at both institutions), yet the prestige of UChicago is nowhere close to that of Harvard. That's due to a number of factors, but as a correlation (can't quite say causation in either direction) the acceptance rate at UChicago used to be immensely higher - it plummeted down to the teens in 2010, but it was roughly ~30%+ for all preceding years. Hence, while the education you might receive upon admission might be comparable to or even arguably better in certain ways than an education at Harvard, admission itself demonstrated very little. And that's the problem - for applying to the Thiel Fellowship, for instance, being an undergrad at Harvard undoubtedly helped me far more than would have being an undergrad at UChicago. Like Thiel said, most of the finalists came from Ivy League and prestigious name institutions - Harvard, Yale, Princeton, MIT, Stanford...not a single one from UChicago.
The exclusivity is, quite honestly, the main thing.
Addendum - though admittedly, the real awesomeness that lies in going to Harvard manifests itself in the peer environment. Contrary to popular belief, pretty much everyone who goes here is social, super chill, extremely nice, and radically tolerant - on top of being insanely intelligent and supremely accomplished. It just inspires you to do things...like drop out and move to California to make it big.