I believe that having more money to spend allows you to pick the cream of available teachers, better class sizes, better facilities and an atmosphere more conducive to learning. This is mirrored by generally better exam results (Is this the part that needs evidence Jack?)Error! Assumes facts not in evidence.
(Most private schools will have some form of selection policy, and having high academic potential will tend to feature rather heavily in such a policy.)
Regarding the selection policy, I had always assumed that selection was divided;- partly by ability and partly by having enough money to buy your way in!
I went to a crap comprehensive and my brother went to a private school. I was at least as intelligent as him. I got a thoroughly crap education, and my brother got a good education with salaries to match!
Of course there are some state schools that can beat some private schools, but generally they are just not. How can all of the many advantages of private schools not make a difference?