I teach in public schools and see a few more differences than what are listed in previous posts. First public schools have to take every child that lives in that school's district, private schools do not have districts they draw from. Second, private schools do not have to follow the same mandates from government that public schools do. Private schools do not have to adhere to No Child Left Behind mandates so they are free to teach what they want when and how they want.
Make private schools play by the same rules as public schools and I think we'll see that private school really aren't that much better than public schools.
Learning must be a three prong approach. The student, parents, and teachers must work together to accomplish any learning what so ever. At private schools parents are more engaged in the child's education because they directly pay for it. At public schools, many parents see school as a form of free babysitting for them, therefore they are not as engaged in the child's education.