I think it's less about academic outcomes, and more about opportunities. There was a study that recently released it's findings that found the educational outcomes from private and public did not differ significantly. Which therefore suggests that how well a child does in school, is an individual thing and not dependent on type of schooling.
However, some private schools offer opportunities that public schools don't. Such as the variety of extra-curricular activities/clubs a child can be involved in, and it can also offer the 'who you know' component as children get to the age they're leaving school for higher education/career options.
But on the flip side, if it's extra-curricular activities you're after, it would probably would be cheaper and more beneficial to opt for a public school and source out of school activities (which is what I do). And the 'who you know' aspect may very well not be applicable to the private school you chose (really depends if it's an "elite" school or not), or you might still meet people in a public setting.
I think picking a school that you LIKE is the most important thing. What are the core values of the school? What are the teaching staff and senior staff like? What's the school community as a whole like? With the right home environment and support, a child can bloom in any school (provided the school is not a complete crash and burn type - which can happen in both public and private settings). It can't simply just be a case of private vs public, in my opinion.