This is a purely theoretical question. Most of us to have our children taught in private schools is unaffordable. I personally have heard only some uncertain information on Hungarian private schools or on luxurious Swiss boarding schools prepared for the children of millionaires.
Having taught at both wealthy private schools, public schools both rich and poor, and in correctional schools I do know one thing, kids are kids. I can also say that the most difficult issue in kids for me to deal with were the ones coming from wealthy families.
What I mean is that there are going to be classroom issues in all economic levels, and kids will be kids as stated. For me, and (it was an issue of mine) not the kids, there was just something different about being entitled that rubbed me the wrong way, I would rather have a student try and attack me than watch that attitude representing "the I know you can't do anything to me".
It may appear that private school teachers care more but please consider this. With much smaller class size the teacher can spend more time with each student. The children going to private school have all the advantages such as home computer, books, private tutor, best of food, and live a very privileged life. I would argue is just appearance and not real. I have worked, associated, and married two teachers in my 20 years and can count on my two hands the number of teachers caring more about the wealth of the family behind the child. One the other hand I know far too many who care about their white kids than they do all the others.
I became a teacher when I was 30 years old for one reason, and that is and still is the belief that teaching the most sacred profession to be in. The trust that families put in us is both a huge responsibility and a great honor. I believe the vast majority of teachers feel the same way.
In private schools they receive and return you back to home with the school-bus, while accommodations are more luxury than public schools. Another benefit is that you can meet there the future prime ministers of your country, provided that is a small one, like Greece (here all pm's are from a specific private school, is this an explanation for Greece's current disaster? Probably yes!). I graduated from a public school and I am proud for it!
Private schools CAN be better, because they have the resources from tuition fees and the ability to pay for better teachers. Of course it depends on HOW they use the extra money.
I am a graduate of the private school that Demetris is blaming for the current political crisis! However, we were privileged to have excellent teachers and most of the graduates (besides the politicians) I know did well after school. This particular school was a continuation of a school in Smyrne that was abandoned due to the Greek genocide in 1922. All Greek schools in Smyrne, part of the Ottoman Empire, were private and this tradition continued with the new school built in Greece.