I would approach them both the same way. Be respectful to everyone. Be yourself. You'll probably see new things and deal with new experiences that you didn't have to deal with in public school. But remember at the end of the day they are still people just like you, so personalities probably wont be as different as you would expect.
I've never attended a private school, but when I was in high school I was pretty popular. Not the kind of popular that you have to try and act a certain way or only associate with a certain group of people in order to obtain. I just treated everyone the same. I hung out with the "cool" kids, shared some good laughs with the so-called "nerds", I played sports so I knew some of the "jocks", and I was respectful to the girls so they were mostly cool with me too. I got good grades as well. And by being cool with everyone and trying to find commonalities with everyone I came across, I ended up naturally making some really good friends, while still being on good terms with everyone I knew. And if I didn't know someone, I just didn't know them, but because I had a good rep with a lot of people, they most likely knew me and I never really had an issue with anyone else, except ofcorse if something caused us to have a disagreement.
Really to sum it all up is just remember the fact that you are still dealing with human beings, and no matter how much someone tries to hide it, everyone has different things and interests about them that may not seem "popular" but are actually quite common. Just find the common ground with everyone you happen to have a conversation with. You don't have to go out seeking friends. But just by you being the new kid, people will start to approach you eventually, and if you just be respectful of others, even if they differ from you, you'll make friends before you know it