I teach at a private boarding school. All of my experience is in boarding schools as I have never taught in a public nor private day school. Everything I have to say is my experience from the four boarding schools where I have worked:
--I started at 23 right out of college. We do not have teaching degrees, but that does not mean our teachers are not excellent and devoted individuals. If one isn't a good teacher, s/he will get canned within a year or two.
--Private schools typically pay less, but boarding schools provide on campus housing (a nice perk).
--Public school teachers are 'graded' on how well their students score on standardized tests. My students do not take state standardized tests and I have the independence to teach however and whatever I want in class.
--Discipline is not really an issue in private schools. Students don't get into a lot of trouble because we kick them out quickly and parents don't want to pay tons of money and then have a kid kicked out. What I've heard about lack of discipline in public schools is why I will not work in a public school. Teenagers are still teenagers and will do stupid things, but it's not a big issue for us.
--That said, I think parents are worse here. We have way too many helicopter parents who have to know every detail of their children's lives and have to have an explanation for every bad mark. They think paying expensive tuition gives them free reign to 'run the school.'
--Class sizes are much smaller in private schools. My average class size for the last 10 years has been about 10 students. Also, I only teach four periods per day instead of six or seven like in public schools, which means I have more planning/prep time during they day.
--I don't know about day schools, but boarding school teachers have to put in a lot of extra hours. We are expected to coach in the afternoons, advise a group of 5-7 students, do 1-2 weeknights of duty in a dorm every week (glorified babysitting), and run a lot of student activities once every three or four weekends. Those 'extras' are a small price to pay for free housing, in my opinion.
I'll probably be boarding school teacher for life because the pros far outweigh the cons.