I can answer this from a personal perspective. I attended a very small Catholic school which even had several classes in one room and nuns were the teachers. When my parents and grandparents moved from Wisconsin to McAllen, Texas in 1960, I entered a public junior high school. Practically that first entire year was a repeat of what I had already learned. In fact, my science teacher would grade my papers first and after my usual 100 score, he would let me help him grade the other student's papers. The next year in high school, they offered major work classes in some subjects which were a bit more challenging than the regular classes. I am no genius by any means...but certainly noticed the differences. The McAllen school system was rated highly at the time...so it was not a factor of it being a bad school.
A generation later, my niece attended several private religious based schools and I think that her education far outshined the public schools teachings at the time. In fact, she was in public school and we (her family) thought that the education was lacking. They had among other things "Fun Friday" which was primarily a play day. And this was in a top rated elementary school! It was at that time, (I think it was 3rd grade) that we had her moved to a private Christian based school.
In addition to receiving a better education, things like respect towards others and manners were taught. Proper clothing attire was also expected. Learning things like those are also valuable. It creates a basis of how to interact with people and sets the stage for success in life.
All in all...my vote would go to private schooling over public. That is sad to say because our taxes pay for our public schooling and we should be able to expect that our children learn many important things there in addition to what they learn at home from their families. Not everyone can pay the additional costs of private schooling.