As everyone agrees it is hard to generalize, but we are trying to anyway. As to eprsonal treatment I found somewhat the opposite of the observation above when exploring where tos end my kids. I.e. in private schools locally, everyone was treated the same, as the democratic philosophy of the schools prevented recognizing that some kids are brighter than others. In public schools that is blatantly obvious so some of them separate out the brighter kids for special programs, not always available in private schools.
Although private schools often had better teachers and a more supportoive environment overall, the cost was enormously out of whack with the relative academic value. I.e. with the cost at zero at a public school and good academics if you look for it, as opposed to cost in the range of $10,000 or up per year in private high school and marginally better although more uniform academics, you really have to be highly in favor of private instruction to think it is worth it.
The main feature that decided me was safety. They had trouble in public school with kids bringing weapons to school, and that did not happen in private school. I.e. if a private school kid steps out of line they can ask him to leave much easier than in public school. So in the end I sacrificed big time financially to send my kids where they would not get stabbed, and would be also taught somewhat better.
The difference for college is much bigger. Just compare the course descriptions at a school like Stanford or Harvard with those at a big state school. Those major private schools are like graduate schools at other places.
E.g. in math at a big state school most kids will take a typical calc course like one from Thomas or Stewart, while at a school like Stanford the brightest kids are more likely to get a course from Apostol.
On the other hand not all private colleges are the same in quality but they all cost almost the same. So at the college level, you pay a huge premium to go to even a mediocre private school, while a top quality school like Stanford is a much better bargain for almost the same money.
Now again, at some state schools, they have an honors program, and if you get in the section that is taught from Spivak or Apostol, they will be happier to have you there and will try to treat you better than at Stanford.
The biggest difference is in the depth of the talent pool in the student body, and in the level of standards allowed by the school. I.e. at top private schools, students are brighter and harder working on average, and the professors are allowed to expect top performance from them. Students are also more inspired to achieve highly by their peers.
At state schools, the professors have to accomodfate more average students and cannot afford to flunk everyone out just because they do not want to work hard, so the student with a good work ethic has to look harder to find a suitable course to take.
None of this prevents hard working students from getting superb training at public schools if they try hard enough. Excellent students come from everywhere and excellent teachers are found everywhere, but the density of them is higher at the best private schools in general.