I myself have attended both types of schools, as well as other types of schools, both public and private.
The BOTTOM line is:
Even with "catholic" private schools, there is a WIDE spectrum of differences in EACH school. It is not all the same.
And this is the same per public schools, of course.
The other bottom line is: going to a private/Catholic school or public, does NOT guarantee... whether a child will grow up better or not, or get into a better college or not, once they graduate high school. IF the parent can keep affording, the private school tuition by high school.
Some private schools I know, have awful kids. Not academically, but in thinking and attitude. Some have nice kids. At one religious private school I went to... sure the kids were "academically" smart and privileged in that sense. But, they were just... terrible in character. It was so obvious, to other kids. So much drugs and drinking and the opposite gender sneaking on campus and the girls making out behind hidden areas etc.
I also have seen some public school classmates, excelling, and getting into Ivy League universities, after high school. Harvard.
I have also seen, classmates who, after middle school, HAD to go to public school, because their parents simply could not, keep affording to pay the private school tuition. Even with financial aid. Then, they had a harder time, financing college, or could not go. After all this prior years, at a private school.
The answer, is not clear cut.
It also depends on, how a child is guided/raised, at home.
It is a MYTH... that kids rebel after Catholic school.
Kids rebel, anywhere, no matter what school.
BUT... if the school is a girls-only or boys-only school... then, to me, and for me, I would not send my child to a school like that. Because it is too, unnatural, socially.