It's been mentioned a few times already but it's important to note that not all state schools are the same as each other, but ALSO that not all private schools are equal either.
There are private schools that are religious based, that may or may not also be academically selective.
There are academically selective private schools.
There are sport-selective private schools.
There are more single sex private schools (this matters IMO), although they aren't all single sex by a long stretch.
And that's before we go anywhere near public schools!
Because they're not all created equal, people's experiences are not necessarily "typical" of either type of school. So it's inappropriate to say "all private schools" or "all state schools" - and when it comes down to the individuals who WORK in those schools then it becomes even more inappropriate to over-generalise.
I only have personal experience of one private, academically selective school, and very close vicarious experience of another.
I went to a state primary school. I know what people are saying about the mix of the two types of school, because I experienced (and, I believe, benefited from) it.
State primary - mix of all types of people: council, very poor, middle class, well off, working class. As a result I had friends from different social classes and from different financial backgrounds. It didn't matter to us at that age, we were friends because of the people they were.
Private secondary - definitely less of a mix. Of course all had to have money, most were middle class (not all), but some were poorer than others (me, for e.g.) I adapted. I still had friends from primary school.
What it gave me, that I felt a lot of my secondary peers didn't have, was understanding of other people's way of life. That NOT every family had 2 cars (or even 1!). That NOT every family could afford to eat at posh restaurants. That NOT every family was able to buy new clothes - all these things that broadened my social spectrum. Some of my secondary peers had been in private education their whole life, and had never come across these differences - so had less understanding.
Not necessarily less empathy (although, y'know, teens...) but just no experience. And less adaptability to different social scenarios.
Of course you can learn this after school - but while at school, it's not so easy.
I did the same for my boys - put them into state primary in Australia - but if DS1 hadn't got into the academically selective state high school programme he did, he would have gone to a private secondary school, mostly to keep him away from the awful awful local high school, which suffers from not only a very poor behaviour record, but has some lousy teachers as well. You can tell they're lousy when a student (friend of mine's daughter) can take the prize in maths for her year with a score of 47%. The teacher who came the next year, as a casual/temp teacher, spent the first half of the year bringing them all up to where they should have been by the end of the previous year - they were 1.5 years behind
.
I built resilience through being the clever clogs at state primary (and bullied for it) and the poor girl at secondary private (not really bullied for it but I had to learn to deal with being "different").