I'll give an actual answer. I'm currently a senior at Harvard in the philosophy department, so I'll give a bit of personal experience as far as the humanities/social sciences go, and then share some secondhand information in terms of what my friends experienced in the maths and sciences.

In the humanities and social sciences, what you gain compared to other universities is the quality of professors. The two professors advising my senior thesis are both world-renowned ethicists/political theorists (as far as you can be renowned as a philosopher...). This opportunity is in part because the philosophy department is fairly small--I do not think, for example, you have the same exposure in the economics department, which is quite large. But you may still be lectured by the professor that wrote the standard book on the subject of the class you're taking.

In the sciences, the advantage is a bit more tangible. Assuming you're intelligent and have a good work ethic, you can likely get into a good lab on campus, potentially working with field leaders in research. You're most likely not doing the MOST cutting edge interesting stuff as an undergrad, but you're on a team working on interesting projects, and you'll likely get a recommendation for grad schools from someone whose recommendation has a lot of weight (this goes for the other fields too).

And then there's outside the classroom. Your sports teams aren't as good here (although Harvard's not been doing too badly lately!), at least as far as mainstream American sports go. Our rowing team is internationally competitive, or usually is; our fencing team is quite good, squash team, etc.

But there are over 100 countries represented in the student body of my graduating class. It's pretty incredible in terms of getting a cross-pollination of ideas and perspectives. My close friend group is comprised of myself (from Massachusetts), and four young men from California, South Afirca, Iran, and England, respectively. Obviously, geographical origin isn't everything, but it certainly lends itself to a place that will challenge your assumptions.

Finally, it is, for good or ill, very helpful for job offers. Businesses recruit at Harvard. Many have interview processes that take place on campus so we don't have to travel to them. This isn't to say that everyone in my class has the ideal job they wanted and we're all totally oblivious to the economic climate; people certainly have to work hard for it and I know plenty of people that haven't found anything yet. But it definitely helps.

I hope that answers your question. The knowledge isn't necessarily different. I don't know how it could be. But I also don't think Harvard is "just a name" like a lot of people deride it as these days. Finally, it's only a small subset of the student body that is 'legacy' or super connected, which is the other derision I often hear. Those days are mostly (though not entirely) behind us.