I happen to have done both - I went to a medium-sized state university undergrad (Go Catamounts!), and an internationally recognized university for grad school (Go Irish!). From a student's perspective, I definitely saw a disparity in resources, which would be expected. The quality of professors was fairly comparably, somewhat to my surprise. While the more prestigious school tended to have more professors with Ivy League pedigrees and stronger research backgrounds, the professors at the state school were, on average, as good at actually teaching. Sometimes better, in fact. The main difference was the quality of the average student. The more selective school had a larger percentage of highly motivated, academically talented students, although the best students at both schools were comparable. There were just less of the "top tier" students at my undergrad university.

Having a recognizable name on your resume does open some doors, especially when you are first entering the job market. As you build you own reputation and work history, it matters less and less, but I do think you still get a little "bonus" on your resume from a prestigious university. I'm not so sure it is enough to justify the huge difference in tuition as an undergraduate, though.