Private schools in the U.S. often do not have as many resources as public schools. The curriculum they teach is typically better because they do not have to make the compromises public schools do, but it is often too expensive for them to provide the "extra"s that public schools can offer, like special education, solid sports programs, the variety of music education, the scope of science classes, any sort of student counseling, and many other things. Most of these are beyond the reach of all but the most lucrative private schools which are surpassing difficult to afford (think $12,000 to $25,000 per year for G9-G12). Even the teachers at private schools are paid worse than at public schools. However, you get teachers who are happier to not have to teach any but the brightest and most willing students and are willing to accept lower pay to do so.
My children both attended private schools for a couple of years. The first school was possibly the best in the city, curriculum-wise. When we found that my daughter had a bit of trouble with reading and was falling behind, they had nothing for her except to tell us to seek private tutoring. We tried another private, all-girls school for her and had only a little better success. When my son, on the other hand, progressed so rapidly he out-paced the class, again they had had nothing for him but to advance him long before he was emotionally ready. Therefore, he languished in boredom like you hear about in public school.
When a pay cut came along for me in the downturn, I had was forced to move them both to public school, a solution I was already considering for my daughter due to the availability of reading specialists. Both are now flourishing in an environment that has a far greater variety of challenges for my son and the help my daughter needed (she now reads above grade level.) This is certainly not what I thought I'd learn, but there you have it.
Private schools have many trade-offs aside from the additional cost.