I can only speak to areas where I've taught.

Montana doesn't have private schools except in the larger areas. There is no difference in performance between the two at the college level. Not surprisingly, the kids who do best are ones that come from very small schools and communities.

Louisiana has a large number of private schools. Those kids do not perform one lick better than their public school counterparts at the college leve. Parents aren't paying for better education, they're paying for safer environments.