Be aware that NOT all private schools are equal, just as not all public schools are equal.
If you compare the bottom of the private schools with the top of the public schools, you will find the public schools better. However, comparing the top private schools to the top public schools can give you a better perspective.
The top public schools, IMHO, can give a "better" education in SOME areas (but NOT all), and the top private schools can definitely give a "better" education in SOME areas (but NOT all).
What you do get out of the private system is smaller class sizes and more individual attention (hopefully.... NOT ALL PRIVATE SCHOOL TEACHERS are good or motivated to do so, there are bad teachers and BAD private school principals as well, even in the top schools).
The point is, your child will have a different social experience. Yes, most people that go to private schools can afford more... (what do you expect when the parents are paying $20K+/year). It is different. In the public system you see people not having enough money to pay for food at McDonalds, whereas in the private system, the kids are throwing money away (IMHO a really bad habit). Case in point, for school teams, private schools just expect the kids to fork out $100+ for the team uniforms. In public schools, it is a struggle to get kids to pay for a $30 team uniform.