Many years ago...
we spent a couple of years trying to develop a business to give school presentations, mostly to 5th graders. We did both public and private schools in Palm Beach County, Fl. What a difference between the two types of schools! In general, the private school kids were orderly, organized, and listened. They asked questions that were reasonably well thought out. In contrast, the public school kids were rambunctious and kept asking the same questions over and over, since they didn't listen to what their classmates had to say. That year or two convinced me of two things: one is that if I ever had kids, I would try to send them to a private school, and the other is that if I ever had to work at a school, I would try to work for a private school. My feeling is that the families who go to private schools are kind of "self selecting". Because there are fees, and it takes more effort to enroll, my guess is that many (not ALL, of course) parents who take that extra trouble probably care more about education. I am sure there are many parents who care about education and can't afford private schools, but I am talking in generalities. Also, a big part of it might be that private schools can throw out troublemakers. Also, private school teachers may not be quite as regimented, depending on the school. If you search some of the old threads that Lauren posted a year or two ago about trying to make classes more interesting for the students, and how it was quashed as not needed, and a luxury, it will probably turn you off wanting to deal with the established school system in most areas. Whatever the difference, I would look into private schools, even though I hear they pay less.
I only know teachers who have taught in public schools, so I base this only on my presentation experience, and what I have read. But remember - that was over 20 years ago that we did this. I suspect the difference may be more pronounced now then it was back then. Maybe talk to some private school teachers and see what they think.