My guess is that it pays more. The public school has great difficulty ejecting persistently misbehaving students, unlike the private school.
The schoolmarm and schoolmaster of old put up with considerably less crap, but also earned considerably less, too. It’s a matter of choosing between the cake and eating it.
“unlike the private school.”
I went to University School in Shaker Heights, Ohio. Mr. Cramer was the school disciplinarian. You did not want to get on his bad side!
I had a friend who taught in a public school. A punk came after him, and he shoved the kid away. He was fired the next morning, and that was 40 years ago!