The main difference between public schools and private schools is that public schools are government-run, while private schools operate on funds from tuition and private donations. Another big difference is that public schools usually cost less, particularly if you're a resident of the state where the school is located.
Beyond that, it's difficult to draw general distinctions. Private schools can be small or large, highly selective or not so selective. Some state schools are very small, and are attended mostly by students who live in the area. Others are large, flagship schools that are widely known, and attract applicants from across the country.
While you're weighing options, consider these factors: