The entry requirements are very different - I have one in university and one applying this year, both in the US. I went to a UK university, and DH has been to both US and UK.
It is interesting looking at the acceptance rate for the top US universities. They take 5 to 10% of Regular Decision applicants, but a higher percentage (often double) of Early Decision applicants.
You can generally only apply to one private university ED, and if you receive an offer then you are committed to going there.
If you don't get an offer in your ED application, then you're free to apply to as many as you like Regular Decision, but your chances are generally going to be lower.
There's also Early Action application - I don't recall exactly how that works.
DD's college accepts fewer than 10% of Regular Decision applicants, which I think is a lower proportion of applicants than Oxbridge, and it's not Ivy League.
Having a high SAT score for most of those top universities will help you get in, but often you're up against at least 10 other applicants who all have equally high scores, so the challenge is to differentiate yourself some other way both academically and in terms of character.