I know that they take extracurricular activities into consideration, which I have a feeling will be more numerous and "impressive" in private school students.

"Ooh, I see that you were in the debating society, and played the violin..."

I think that there's an element of self-perpetualism about things. Academia is is packed with private school alumni, who will tend to seek out entrants similar to themselves.

I did get the impression when I went to my Cambridge interview that I was being tested to see if I was "the right sort". Including questions about my parents' economic situation, which unnerved me somewhat.