The inherent difference between public and private schools (other than cost)
Private schools - try to give students (most of whom want to be there) an education
Public schools - try to prepare students (many of whom don't want to be there) for state tests
Really though, it depends on area. In my area, the private schools are much better than the public schools. The teachers are better at the private schools in my area because, even though they are paid less, they are treated much better. Smaller class size, fewer classes, etc.
In regards to expelling students...it is nearly impossible for a public school to flat-out expel a student because of laws. Unless a student violates the safe schools act, the public school must provide him services. So while you may think a kid got kicked out, he really is still going to that school in some form.
Many private schools kick out kids who don't make the grades or otherwise bring trouble. The good private schools don't worry about that kid's tuition dollars. They will get the money elsewhere.
Also, most public schools with advanced degress have them in education, which is utterly worthless (except in the sense that it allows them to make more money). Teachers in private schools with advanced degrees tend to have them in their subject area. Since you can't teach someone to teach, seems like it would be much better to have more knowledge in your subject area.