First let me begin by saying that my entire life I was educated in a private school system. But for many years, I have been employed in the public school system. In terms of better teachers or higher pay, the private school teachers were drastically paid less. The public school receives funding from the government and the majority of our taxes actually go to them. If you teach in a large city, your paycheck will be substantially higher than if you teach in the outer suburbs. More people pay more taxes, therefore your school will be receiving more money.
As for the quality of teachers, they are basically the same. I will say that the public school has more computers and have more updated books in their curriculum since they can afford them. The private school children that have transferred into our public schools are usually one year behind in their education. When it comes to safety, my daughter went to a private high school and in her first year there was a girl with a gun threatening to kill some other girl. The reason the gun went undetected is because private schools don't have the security or screening measures in place that a public school does. Public schools are often overcrowded and the teacher to child ratio is worse than in a private school. But it all comes down to the child.
A child will succeed if a child wants to. Bottom line. In terms of education and teachers, a private school is no better than a public school. In the end, they all get the same diploma.