I went to both public and private school, and I'm mixed on whether the teachers themselves were that different. What absolutely made a difference were the stakes—if you fucked up in private school, there were actual consequences. Generally, the kids at the private school had parents who cared, either because they invested effort into their kids being smart enough to succeed there or because they invested a lot of money each year into their kids' private educations. Teachers didn't have to tiptoe around feral retards who might create a crabs-in-the-bucket mentality, since they could just get kicked out of the school entirely. And the private school knew the academic and professional success of its students was directly correlated with its ability to make money, whereas unionized public school teachers and admins can literally rape kids and get away with it because there are no market forces at play.
One of the biggest differences was that you could get alienated by your peers in a good private school for detracting from their educational success. Peer expectations (or the lack thereof) keep Baltimore retarded. But in a private school, elevated peer expectations push students to perform better. Again, it's culture. Unfortunately we're not allowed to speak seriously in public about how certain cultures are incompatible with a civilized society that emphasizes education.