As someone said, one major problem are the teacher's unions. Unions can be a good idea, but not to the extent these have gotten. Those need to be fixed, most definitly. You can hardly fire incompetent teachers. However, that is not the only problem, and some of the problems are quite intrinsic to the concept of a mass-education.
Private schools do not have to accept everyone; public schools do. Public schools are not allowed to cherrypick students that are easier to deal with, smarter, come from better homes and areas. Many, but not all, private schools do this. Public schools must also adhere to rights of teachers and students, while, to a lesser extent, private schools do. Mass education comes with problems of a mass education--you have to deal with all kinds and you cannot simply say: no, you can't come here. Private schools typically are smaller and service fewer individuals, and they like it this way, since it is one factor that helps to keep their stats up compared to public schools. As an example, when taking standardized tests, many of the "retarded," slow, or problem children have to take the same assessments as the good or average kids, therefore, the test results go down.
Another problem is that problems in results are not always the fault of the teachers or the schools, but the problems of the students and the parents. I ahve said time and time again that many kids simply do not study. Half of the kids in my biology class in COLLEGE--people 19-20 years old, fail consistantly, the basic 103 level multiple choice questions. This is not the fault of the institution. They are just stupid, careless, or ignorant to the subject's importance in life. The parents are also to blame. Many do not have time for their children's needs in school, and in highschool, parent identification with and help for the child declines compared to grammar school. This is a problem.
Then you also have the case where parents are incapable of doing the type of work modern students have, since they didn't have it. It is even changing in my life. My school never taught what it's teaching now in 7th grade. I didn't even have physics or trig.
The best teachers cannot help massive of kids who don't care, don't want to learn, and get no help at home. This can easily be avoided or minimized at private instititutions, since the people who send their kids to these schools oft pay big bucks and care more, and they damn sure make their kids work because of the $$$. Not everyone can compete with this mentality.
Underspending is a problem in some areas, and this combines with administrative and municipal overburden. However, this is not the only cause of inferior scores and results; misapplication and misappropriation of funds is a problem. Another problem is that people have far too much say in what goes on in schools. MOre power ought to go to educational technorats in the field and the classroom instead of allowing parent/community or far-flung officials to say what ought be done and then ride off into the sunset patting themselves on the back. IF you want to see an example of this problem, look at Kansas and Dover PA.