I would say that the private schools are teaching material earlier than their public school counter parts. I went to a Catholic private school from K- 7th grade. I found myself learning much of the same material when I was in 8-11th grade as I had when I was in 7th grade and under. Of course it wasn’t like that in every subject but in math and English it sure was.
Now whether or not the students in private schools are getting better grades just comes down to the particular school you are talking about, surely not all Catholic and private schools have better performing students than public schools.
I think that schools run by the monastic orders (Benedictine, Cistercian, Jesuit) tend to be more exclusive/rigorous than parish schools (run by the diocese). But that it strictly an anecdotal observation.
I think you’re drawing a false dichotomy, there. “Run by a monastic order” does not necessarily mean “run by a monastery”. Most Catholic schools are run by the members of some particular order under the jurisdiction of the diocese.