My daughter attends one of the top private schools in the nation. My wife and I both attended public schools and my wife has taught in both public and private schools.
At my daughter's school, 100% of all graduates go on to attend a four year college. Many are National Merit Finalists and many end up in the Ivy League.
So this school is obviously doing many things right. Some of this has to do with the financial advantages of a school that charges a high tuition and therefore has all the resources it needs. But most of the things they get right (and that public schools get wrong) have nothing to do with money.
Here are things that my daughter's school does that public schools could do to improve with little or no extra money:
1) Students have PE every day. PE time starts with laps around the gym. The studies cited above are correct. Students perform better academically when they are in shape.
2) Students are expected to behave. If they don't they are kicked out. Period. End of story. You can't let one disruptive kid ruin the learning environment for everyone. If you don't remove behavior problems then the teacher ends up spending all of her time with that one brat. And the rest of the class suffers for it.
3) Teachers are expected to teach. If they don't they are fired. But in private school, they hardly ever need to fire anyone because it's a great place to work (see #2). In public school, teachers eventually stop caring because they are unable to do their jobs. And why should they care? They won't be fired.
4) Teachers are hired based on a proven ability to teach. Just because you have a teaching certificate doesn't mean you are any good at teaching. To get a job in the private school, candidates have to actually teach a lesson in front of a real class. The best teacher gets the job. Public schools never do this. Mostly, if you know the principal and have a teaching certificate, you get the job.
5) Limit the number of standardized tests. In public school, the kids are constantly taking standardized tests. This takes time away from teaching and generally does not help with the actual evaluation of the student. The tests are more about evaluating the teacher, which you wouldn't have to worry about if you followed #3 and #4.
6) Group kids by ability. Public schools rarely do this any more in the lower grades. It doesn't make sense to hold some kids back because others need to catch up.
7) Bring back music classes. My daughter has music twice a week. The benefits have already been stated.
8) Recognize reading disorders early. It is estimated that as much as 20% of students have some form of dyslexia. If it is severe, these students are placed in special ed, never to return to a regular classroom. The ironic thing is that dyslexic students tend to have much higher IQ scores than the general population. If the dyslexia is not severe, the students are left to flounder and just considered stupid. But dyslexia can be treated, especially if it is caught early. Those with the disorder who overcome it tend to be very successful later in life.
A public school will do everything it can to stop a child from being labeled as dyslexic. This is because it is very expensive to treat and the school would be compelled by law to offer extra help. But it's much more expensive in the long run to let the child suffer. It's cheaper to treat dyslexia for a couple of years than to place the child in special ed for the rest of his school career.