I used to teach at a private school. I LOVED my job. Now I teach at a public school, and, well, um... not loving it so much.
The pros of a private school are basically that you spend more time teaching and less time disciplining. That's not to say that private school kids never misbehave, but if kids at my old school behaved the way my public school students behave on a daily basis, they were thrown out of school and not invited to return. In addition, the private school I taught at had a fairly tough entrence exam, so we didn't have the kinds of discipline problems that come from students misbehaving out of a simple lack of academic interest or ability. Finally, since parents are paying to send their kids to school, the parents who send their kids to private school tend to be educated and involved - all of which adds up to good students. I was teaching the kids who were the most fun to teach, no doubt, because they liked learning and came from backgrounds that supported education.
The problem with private schools is basically the fact that private schools are less democratic. I think the work of public schools is important - all children, bright or not, well-mannered or not, deserve an education.
On a less idealistic level, the other problem with private school is money. Private schools generally pay significanly less than public schools do.