I've attended both. There's no question that the results of most private school educations are superior.
Having said that, there are arguments about why.
PS teachers I know say that private schools can cherrypick the best students. Probably true, although most have extensive outreach programs and scholarships for poor kids.
It's also true that parents who shop around for better schools are probably more engaged, and thus will encourage and work with their kids more than those who just send them off (or not) to whatever's handy or legally required of them. The kids of parents who care and work with them will no doubt have better educational performance, regardless of socioeconomic class.
One huge issue is public mandates. Under the Americans With Disabilities Act, public schools are virtually forbidden from expelling students no matter how badly they behave. There are probably funding strings attached to special needs students, so it's to the PS's advantage to classify them with some kind of learning/behavioral disability, so thus Viola- they are disabled and protected under law, and must be placed in a public school and given all available services. The private schools don't have to put up with that. Screw up, and they kick you out.
Public schools are also required to educate kids with serious mental and physical disabilities, which will inevitably drag down their scoring performance as a whole.
Vouchers are a real Catch 22. Private schools could certainly help shoulder the load and give good education to kids. Unfortunately, since PS's are funded based on body counts, the more who leave, the less they get. It stands to reason that they would oppose vouchers, because every kid that leaves the public school system takes state funding money with them. Unfortunately too, the central city PS systems don't really acknowledge that the funding isn't going to private schools, but to charters and the suburbs who are gaining population from dispersing of inner city poverty to suburbs.
I don't know the answer. It just stands to reason for me that if you have fewer customers to serve, you can spend more time and effort on each one. If we could get the money issues resolved, it could be a win-win.