Private schools/parochial schools, can accept students from anywhere regardless of where they live, that's not the case for public schools?

Public schools get funding from the state.  Private schools do not.

The quality of the student body is better at private schools than at public ones.

In my opinion, the reason why private/parochial schools are more successful in sports than public schools is the support system. Case in point, my sister was 4'11" and 98 pounds, and went to a private, all-girls high school. She was a year older than her neighbor, who was the star running back at the local public high school. In the summer of 2007, my sister was entering her senior year of high school with no football experience, while the neighbor was entering his junior year of high school after earning all county football honors as a sophomore in 2006. My sister bought a new pair of size 8 Adidas Falcon W sneakers that summer (which the private high school allowed her to wear after school every day), while the public high school did not provide the neighbor with the best shoes/equipment that year. As a result, the neighbor struggled and took a step back during his junior season, while my sister took full advantage and beat him out for the 2007 all county football honors.

I have found public schools have a much better sports department over private HS. 

There are no residency restrictions for most private schools. Students can come from anywhere, and transportation is usually not provided (although the public school buses transported to and from my Catholic elementary school when I attended). This absolutely helps these (usually) small schools compete so well in athletics. They can and do recruit athletes to play for them. They can offer scholarships as well. 

i think thats the difference too