This is no surprise, but before people start spouting shit about how useless the private school system is I'd like to add my experience to the discussion. I attended both private and public schools. I can tell you there is a difference between them, but it's not strictly academia. That's more of a byproduct of opportunities rather than just a 'you will get a better education'.

For me the difference was mostly staff as well as the expectation put on students. Staff at public schools were satisfied with students meeting the state average, whereas at the private schools they pushed you to get better at the areas in which you showed interest.

I went from the top of the class at a public school to the bottom of the class at a private school when I transferred. The difference wasn't that the students were smarter, it was that the private school system pushed them harder.

I was never given opportunity to participate in extra curricular stuff or programs which offered greater learning at the public schools. I would breeze through homework etc at the public school to a standard which the school was happy with (slightly above state average) then spend the rest of my time bored causing trouble.

At the private school they recognised my abilities in certain fields and pushed me to participate in extra activities, work harder on certain things etc, and before long I was back to the top of the class. However this time I was sitting at a much higher standard.

I am confident if I had stayed at the public school I would have been complacent with mediocre grades as that is all that was expected. It was teaching me that average is good enough, while the private school taught me that if you want to be good at something you can be with enough work.

tl;dr public school taught me that average is good enough. Private school gave me opportunities and showed me that if you can succeed if you try.