I'll share my anecdote from attending a private catholic school. From the reasons I have heard from my parents and others is that the catholic private schools tend to be higher quality than the public schools without being outrageously expensive like the non religious private schools.

I don't remember seeing any student at the school who was actually religious, they were all there for the perceived better education. The only part that I saw was lacking was sexual education was non existent outside of one lecture about the different kinds of stds and how there is no way to prevent getting them other than to not have sex. The science education was still up to scratch however.