Public high schools are handled and managed by the government. Private schools are owned and managed by private individuals or corporations. From where I am, those who go to public schools do not pay tuition fees. There may be other fees but very minimal. Private schools on the other hand can dictate their own fees but are regulated by the government.
In terms of quality of education, most public schools don't have much.
Being filled to the brim, because it's free, public classrooms have very few teachers to handle a lot of students. The reverse is true with private schools so you can expect better teaching there. The problem here however is that public schools pay much higher than private schools so the tendency is for fresh graduate teachers to get experience in the private school then move to the public school for good. This complicates matters because with a continuous change of faculty, private schools are having a hard time maintaining instructional quality. Add to that fact that more and more teaching graduates go to work as maids or caregivers abroad.
Discipline is generally stricter in private schools with lesser students to worry about. In the public, students usually can get away with almost anything, even attendance.
Curriculum is the same, but the manner of teaching varies. Private schools who are better equipped can of course teach better.
Too many points of comparison to include in this answer. I feel I can write a paper about it. If you're looking for a place for your kid, go to a private school if you can afford it. If not then no choice.