I am a teacher of about 20 years and have taught in both private (single sex girls and boys up to 13+) and also state schools (juniors).
My findings are this -
Children in the private sector have one teacher to 12-20 kids plus specialist teachers for certain subjects. They are also often tutored outside school and usually have access to amazing facilities within the school.
State sector kids are taught by one (usually very stressed) teacher to 30 kids, who teach everything (and do a bloody good job). Limited access to resources but if the school is good, the kids can do very well.
Teachers are good/not so good in both types of schools - no difference at all.
However, in affluent areas, state school parents can access the same as private school parents outside of school (eg tutoring, clubs etc etc) which makes up for any shortfall at school, if that is what is important for you as a family.
My 2 both go to the local state schools, which have been brilliant and both kids have done very well. But yes, their schools are good/outstanding and we live in a fairly affluent area, plus we have given been able to give our kids access to out of school clubs/top up tutoring when needed etc etc plus support them at home.