Private - Public
I think I'll post a few differences between the school I am working at now and the school(s) I was working at last year. My current school is a private school while the previous ones were all public schools which may bear some relation to some differences, but I cannot say for sure in all cases.
Students are generally more polite, though in a polite more than friendly way outside classes at my current school. This, by the way, bears no relation to behaviour in the class, which I find fascinating.
Classes always start with a bow and 'onegai-shimasu' from the students and teacher at the beginning of my current classes. Also, boys are referred to as name + kun suffix (for young men) while girls are referred to as name + san suffix (standard Mr./Ms.). I don't remember this distinction at my old schools so I presume it doesn't exist there.
Also of note is that at my previous schools students could freely walk around the staff room helping to clean at the end of the day or while speaking to teachers, perhaps after introducing themselves at the door. The only time they were not permitted to enter was during exam periods. In contrast, at my newer school students are never allowed into the staff room. If they are looking for a teacher they must knock, announce their class, name and state their business. It's quite a lot stricter.
Finally for today, uniform deserves a note. Whether it's a fact for all private schools I am not sure, but the level of strictness concerning uniform is much higher. All students pretty much have black hair and there are absolutely no loose socks. This is vast contrast to my other schools were many students had dyed hair, super short skirts, loose socks and so on and so on. However, even those students tend to dye their hair back to black during their third and final year when they are applying to universities.

