the short answer is yes, courses are harder at elite schools, but the answer is more complicated than that. When I was there, there were non honors routine math courses at Harvard that were somewhat comparable to those at average universities, but there were also super advanced honors courses at Harvard that were light years ahead of anything at an average school. So the range at Harvard is greater. they do have a routine calculus class for pre med students or people who don't want the hardest math course in the country, but they also have math 55, and there is nothing like this at most schools, even good ones. If you only have a handful of super top students, as is the case at most schools, you cannot offer a whole course aimed just at them,
From my experience as a student at Harvard and a professor at a good average state university, Harvard, MIT, ... are blessed with more well prepared, more motivated, and harder working students than we have at most places. Not necessarily smarter, as I have seen some of the smartest students at state school I have ever met. But they are not challenged as they would be at Harvard.
There are exceptions. When I was a professor at Central Washington state college, I wanted to provide the best opportunity to my top students, so I taught an extra calculus class from Mike Spivak's book to the few who could handle it. I taught it free so my department did not mind that it only had three or four students. We went from that into Spivak's Differential Geometry as a seminar, and those students, and several faculty who attended, learned a lot.
I also taught advanced calc once there from Spivak's Calculus on Manifolds, and although I blew away most of the class, one student who had enrolled during a hiatus from Notre Dame, liked it fine and went back to Notre Dame the next semester. But they never let me teach it at that level again, because for the other students it was inappropriate in the extreme. I once offered a class there, complex analysis, that only enrolled one student, as no others were up for it.
So in any school you may find exceptional classes that some students will benefit from, almost as if they were at a top school, maybe more so since they will be treated very personally.
Many state schools have outstanding professors, e.g. my friend Ted Shifrin at UGA, who previously taught at MIT and Berkeley, and has offered advanced math classes at UGA to top students for years. But they may have to stop offering those classes soon, partly due to lack of demand.The one thing you may find at Harvard, MIT, etc... that you will probably not find at most other schools, is a super advanced class in which the professor who is a world expert in a topic teaches new material he/she actually created, and it just isn't known in that form to the rest of of us.
E.g once in a graduate course on differential topology Raoul Bott wanted to present the proof of the Kodaira vanishing theorem, but he found the proof in Kodaira's book too tedious. I myself had slogged through that proof once in a seminar and I agree. But Bott simply made up another nicer proof. I recall his saying "to read that proof you have to have real stamina!"..pause..."I didn't have that"...great laughter..." so I used the principal bundle". There were Fields medalists sitting in that class with us to hear his lecture that day. This does not frequently happen everywhere I suspect, although even average state schools also have world experts on some topics.
But most of those classes are not for most of us.
Just trying to get a better sense of what this difference entails. 
First, which Calculus book by Spivak are you referring to mathwonk? I'd be interested in checking it out. You've really piqued my curiosity.
