I believe they are morally superior. In as that they are able to permit religious freedom where such in public school is savagely curtailed.
Morality is openly taught in Christian schools where morality is not implied in public schools. The same values taught to previous successful generations are taught in Christian schools where public schools are now havens for communist and socialist indoctrination. Social promotion in public schools is normal, promoting children based on social acceptance where these private school still apply the "achievement" standard. That's what the parents of those children pay for.
And finally, Christian morality wins out over a moral vaccum every time.
This makes them superior to public schools in every sense. Religion has it's place in school, there is no dispute over this, as we have seen the results of banishing religion from public life. This is a Christian country no matter what anyone thinks, the only reason the founding fathers felt morally justified in their actions was because they felt they were endowed with such inalienable rights by their creator.
Public schools seem to more or less want children to forget that. I believe it's simply a process to destroy any sense of such rights and desire for freedom within this nation by raising generations of children who simply do not know that they are entitled to such.
As far as acedemically superior, I'd state in general terms easily. Discipline and respect are essential to academic progress. Usually these schools are a step above in those areas, so the children who attend are already several steps ahead on that alone.
I'm of the school of thought that morality should be taught at home.
As for academics, I can name a few public schools in NYC that outshine almost any other school, private, religious or public in the country.