Public schools are obligated by law to educate all children, so to enroll in a public school you simply register your child by filling out the necessary forms. Public schools must accept any resident student who applies for admission, regardless of sex, race, religious affiliation, economic status or physical or mental handicap. Public schools must also meet state graduation requirements, which vary state by state. Public schools can kick children out if their behavior is too disruptive; but the public school system will usually have in place an alterative school that the child will attend until they are no longer under the care of the public (18 years of age).
Private schools are not obligated by any laws regarding admission. Therefore, private school admission is competitive. Also, private schools are not required to provide educational programs for children with special needs. Private schools are also under no obligation to keep a student enrolled. If a child’s behavior disrupts the school’s milieu, they can be kicked out. Another scenario to keep in mind is that if a child’s academic progress is not acceptable, they may be kicked out as well. Graduation requirements for private schools are decided by each school and are not subject to any state requirements. Many private schools do choose to align themselves with private school associations which will mandate graduation requirements.
The tiny percentage of students in private education systems may very well gain a statistical advantage over their public school counterparts when it comes to SAT scores. This phenomenon is known as cherry-picking. I suspect that if you look at the top ten percent of all students tested, the majority of them will be products of public education.
I'm sure that there are poorly performing public schools out there -- there are also some very good public schools. The same can be said of private schools. Being private doesn't guarantee quality.
Yes, many private schools in my part of the country teach "creation science" and other such religious nonsense. No, our public schools do not take punitive action against private school students. In order to enroll, students must take a standardised exam to determine their grade level. I am not an educator. I am only relating my own, personal anecdotal experiences.
Want a non-anectodal take on the issue?
Read this: <link>