Fwiw, I went to two private schools, 1 public one, oh and one military one.
The worst school, by far, imo was the public one. Even though it was a top tier tech school. Worst equipment, worst setup, dumbest paperwork, etc. Had lots of good profs though, and some REALLY shoddy ones. All my schools had lots of good profs.
Both my private schools were pricey and largely full of well-to-do kids. Think entitlement, expensive cars, "I like to drink, a LOT!!!". Some were really good students, others weren't. The best students I ever had were the hard-working well-to-do students at my second school. My public school, I was utterly appalled by some of the students there [who, I'm sure, if things were merit-based would never have gotten in in the first place], as well as my first school. "Why study when you can party" right??? Especially if it is all on mommy and daddy's pennies anyways/
I think anywhere you go there'll be rich/elitist/snobby/self-centered people, but well-to-do schools do probably have more of them. Entitlement was a big issue at both of my schools, "How DARE you give me a B on this lab report!!!" I think I got cussed out by a student once about that. Even though it was a physics lab, where answers are generally very black or white. Several of my students swore at me actually...
Fwiw, and good luck getting into one, the best school may very well have been the military one. Its not really a fair comparison since that one was ONLY a grad school, but man those people are thorough and have their shit together. Plus they have resources that would make others gasp in ahh. My tech school didn't have squat on the military one. nd its military culture, so you'll see MUCH less bitchiness, entitlement, self-centeredness, "I'm only here to get drunk" etc.
On another note, something I was told way back in the day, and having been to many schools [including just visiting] I really see it more now in hindsight, its very important to actually go and visit any school you are interested, and get a sense of its "feel". "Feel" can very TREMENDOUSLY from one school to another. UK, the University of Kentucky at Lexington, had a very different feel from any of my schools, CU Boulder [Colorado], and OU [Ohio University Athens OH] actually all were kinda similar to each other, well to do rich white athletic/fit people who wanted to have a good time. Very different from the drunken revelry of my undergrad, the "most of us drive BMW's/Lexus/Jaguar our parents bought us when we turned 16 and go skiing every weekend" socially-conscious-volunteering of my 2nd school, or the "people line up 5 or more deep to offer you a job before you graduate, but there are few women and were all basically nerds here" tech school I went to. The military school was, surprise surprise, almost all military officers, plus a few civilians who happened to live in that area.
Alright, just some of my thoughts, hopefully they are helpful for something.
Oh, one more thought, maybe the biggest thing that made me inclined to a private school was avg class size. I knew I would major in math, and didn't wanna end up in some hard class with a prof who didn't care and who didn't know me, and that was too busy for me to ask questions or get help from. Doing the private school, I had small class sizes, and never had issues going to see a prof if need be. To me that was worth a lot.