The only real difference from Public & Private schools is “privilege” which is why narcissism among the faculty of private schools is practically a requirement. All private schools are businesses with the sole purpose of making money, whether it’s a non-profit or not.
"What a lot of people who subscribe to private schools are paying for is not education – or is only partly education. They are buying the “privilege” of making contacts with other privileged people, of keeping to their own kind, and often having better appointed building and facilities and better teachers (because, presumably, the schools can afford to pay them more)."
"Private education as we have known it is on its way out, at both the K-12 and postsecondary levels."
"One can also fairly ask whether U.S. private schools and colleges are really all that different from their public-sector counterparts. In practice, their education-delivery model is practically indistinguishable, save for the accoutrements that the wealthiest of them can buy (trips to faraway lands, nifty technology, tiny classes, etc)."
"In less prosperous schools and colleges, religion may, at day’s end, be the only real difference between public and private – and the return on that investment, while perhaps significant, cannot be easily measured."