As an American I have a lot to say on these issues.........though before you all begin judging, believe me, I'm one of the most anti-American Americans I know, and I'm in NYC.
1. When I've mentioned to people here that I'm even applying to Oxford, people are shocked. The general conception is that while Harvard and Yale are very prestigious, Oxford is just in another class within itself, more than just prestigious....it's like a world class university. Most people aren't aware of the Oxbridge professors being suctioned across the Atlantic, and of the difference in funds available, however, so really Oxford is always considered superior. Maybe sometime in the future these facts may catch up with reputation, and roles may switch - but a 1000 year history of excellence is not easily overcome, and Harvard and Yale were created in America trying to model themselves after Oxbridge, in hopes of attaining a fraction of the presige associated with them. Saying you go to an Ivy League school is great but you can picture overachieving, doing-everything-so-that-they-can-make-an-ivy students with no real talent going there, while at Oxford you picture the future Nobel Prize winners, esp in humanities subjects (probably b/c the funding does influence science/math research here) and the future Charles Dickens'. Stereotypes and reputations and hype abound, but there you have it. Personally I like Oxford, so there ya go.
2. Oxford vs Cambridge - Although I can't answer for the UK, in the US it's definately Cambridge. This, again, has little to do with facts, statistics, or even reality, but reputation. Even in books like "The Great Gatsby" by F. Scott Fitzgerald, representative of America in many ways, Gatsby was an "Oxford man" just b/c that sounds like the most prestigious university you can go to.