I think for undergrad, you can't really beat Oxbridge - the tutorial system has no competition from any available from the top American universities, where class sizes can be huge for some of the most popular courses. However, for graduate studies, it is probably different. I am of course biased though!!!
I am a bit envious of the whole 'liberal arts' set-up though - much as I love PPE, it would be nice to do a bit of English and History at university too. This is something you can do in the States. That said, your specialised knowledge at the end of an Oxbridge course will be far more advanced (I would have thought).
For international reputation, Oxbridge still I think holds a very high position. In the US I would say that Oxford has a definite edge in terms of name recognition - though why I'm not so sure. Maybe just more exposure in the media than Cambridge (from Brideshead to Harry Potter), or maybe Americans sometimes mistake 'Cambridge' as meaning Harvard/MIT.
You've got to remember also that the big American universities have far, far better resources. They can afford the best facilities and can lure the best academics with the best salaries. Given this disadvantage, Oxbridge does pretty well for itself. But if Oxbridge wants to remain at the top of the world rankings in the next century or so, it needs to go some way towards where the American universities are now.