I went to about 20 schools growing up, mix of public & private & christian & even catholic. Generally:
Teachers about the same, some good some bad. Private had some awful ones even in 'top' schools.
Facilities - massive difference in sports facilities. Also a very strong focus on sports in private. For me personally, I was just as happy doing the public school sports for the most part although I did enjoy sailing at a private school which had it's own boat yard etc.
For my kids I'll probably go public, definitely no religious school unless a secular option is provided.
There's a book 'free schools' which sounds very good, author was on conversations with richard fidler. He looked into all the data, came to the conclusion that school does not affect kids grades, socio-economic background and home life is 95% predictor. Rather than provide a better education, all that a private school does is attract those from better backgrounds, hence the better grades. Plus they scoop the cream from the public system with scholarships which further improves their grades. His other conclusion was that every school different, good and bad examples of each, need to take the time to really get to know each option available to you.
www.booktopia.com.au/free-schools-david-gillespie/book/9781742612195.html?source=pla&gclid=CIPeifHo8r4CFcJ8vQodBHIAYg