I don't thinks comparing public and private schools is that simple (or fair) There are lots of different public schools (selective, collegiates, performing arts, sports etc) and lots of different privates (independants, catholic systemic, christian/religious, alternative etc) and all have their pros and cons. The only "simple" comparison is cost - you have to pay to go private and even this varies greatly amongst schools.

I think the more important questions/perspective should be -

what schools are available in your area?
what are the strengths and weaknesses of these schools?
what can they offer your child? subject choice/extra curricular?
where will your child be happiest?
where will your child reach their full potential?

To me the success of a school comes from its staff/teachers (and this can change over time as teachers come and go) No amount of money/buildings can replace teachers who are passionate about their craft and these great teachers are found in public AND private schools. (I work in a public school and resent the common implication that the best teachers work in the private sector)

Talk to other parents and students in your area (a variety) to get a feel for each school and make appointments to visit each as well.

Education is a partnership between you, your child and the school you choose. If you are active in your child's education - they will be fine. This - to me - is far more important than the amount of money you spend. I am of the opinion that a determined child will do well anywhere - with the right support.