Truly, the private school-public school issue is worthy of a book length treatment. However, just a few things to keep in mind:
1) A high portion of the total money in education is spent on hiring staff. Public schools pay most teachers a good middle class wage with good middle class benefits, and private schools usually do not. It is easy to demagogue this issue by attacking the wages of the most senior teachers in the higher paying school districts but that distorts the truth. The typical public school teacher makes a little more than $40,000, and the private school movement is, at its core, aimed at slashing that figure.
2) It is also generally true that private schools get to cherry pick. They have control over who take on as clients. For the most part they don't take on those students who are the most expensive to service, and, for the most part, the most expensive students to teach do not come from families who would go to the effort and personal expense to go to a private school. Public schools, by contrast, must provide education to whoever happens to live within the boundries of the district. You would be shocked at the portion of public school funds that go to special cases. Much of this is good, and it reflects the will of public opinion and American values; some of this is fraudulent and based on highly questionable science -- but almost none of it is truly within the power of public schools to control.
3) Test results are more misleading than not. On the simplest level, you must remember that those students who attend private schools are vastly more likely to have parents who value education. Since parental concern always has been the number one predictor of academic success, one would expect this group to do better in school. Also, the average private school student comes from a wealthier family than the average pulic school student, and socio-economic class is also highly correlated with academic success.
However, those answers are just the standard debating points. The honest truth is that educational statistics have little relation to reality. Don't read the following as an excuse -- just bear with me a moment. Given the number of non-achieving students due to absenteeism, teen pregnancy, substance abuse, parent apathy, etc., it is very difficult to dramatically alter the overall statistics over, let us say, the thousand students in one high school. Classes can be well taught, and individuals can be helped, but there are typically large numbers of students who are beyond reach, at any given time -- and those students have a profound impact on the school's statistics. No matter how much (or how little) effort, money and expertise have been given to the matter, statistical results are demanded, and professional jobs and reputations are on the line... In the real world, this leads to fudged statistics: teaching to tests, easier tests, easy grading of standardized tests, last second grade changes, and pressure on teachers and administrators who do not play along.
This is the real life outcome of school competition, and the results are poisonous to learning. Teaching to tests may be the way for the school (public or private) to survive, but it is not the way to engage students in learning nor to educate a new generation. Nor is it any reliable way for the public to decide where education is working and where it is not.