(Disclaimer - I will speak of the schools here in the US as they are all I know)
It's impossible to lump private schools into one group and equally impossible to lump public schools into one group as well.
To answer the question, I have years with both, and I also have been working for more than 10 years at a public school.
So I definitely identify with the public school identity as far as that goes, but man, public schools really should be broken up into: inner city, suburban, rural at minimum, and with southern, midwest, west coast, east coast, and north/northeast sublevels for each. And then you also have the structural differences between county/parish ran schools and chartered schools.
Private schools can comprise of various religious sects, non-religious, also inner city vs suburban vs rural but to me the political factors on how the private schools can be structure HUGELY affect what the school becomes and how they operate. Much more so than for public schools on average. You will also find spending and funding much more swayed/affected by said politics. There are some mega fancy church schools out there, and then there are some real seedy dumps out there as well that still charge a fortune to attend.
Being in the south, especially in the bigger city than not, private schools are largely baptist/protestant and vastly monoethnic white, usually well-off financially. My experience is similar to how I think of things even to this day - the kids and the adults there are so disconnected from reality living in their sociopolitical bubble. It was hard to get along with people in general as far as finding people to identify with and connect. I also hated how drab the learning experience was, as if the uppers didnt actually care for academics.
The other half of my growing up was in podunk town East Texas. Rural public schools there are not that much better as its still pretty damn monoethnic and ran by the local politics of the town, since everyone goes to just one school. But the best part was the academics were untouched by the local idiots because its a state-mandated curriculum, ran by people who tend to be better experts than the maws and paws of the farmland. So while I still didnt find a lot of people to identify with, I at least enjoyed the classes there.
The school I work at was a school I wish I could have been a student at over the above two. While the demographic remains monoethnic (minority AA) the openness and down-to-earth atmosphere is shared by the majority of the student body and that's what I never experienced from my grade school days where everything was more set in stone as far as what you were expected to be and think like. Leadership here has made my job life wonderful and having the autonomy to tailor the administration to fit the student body as needed is a great side to being chartered, and we of course still follow the state mandated standards as required by law.
Sorry for the long winded answer ^^;