We can look at this question from two different perspectives: That of an individual student, and that of society as a whole.
Private universities tend to be more expensive, but with a few exceptions (the top public schools such as UC Berkeley, Wisconsin-Madison, the University of Michigan) they are generally more prestigious than public universities. The quality of education is widely considered to be superior at private schools, and the opportunities a degree will bring are generally much better as well.
So from an individual student's perspective, it's really a question of quality versus cost: Assuming you can afford the private school at all, it is most likely better; but is it enough better to justify the higher price? The specific results will also depend on what kind of scholarships and financial aid the student is able to get, as well as what field they want to go into (some universities are very good in some fields but not very good in others). As a general rule, go to the most prestigious school that accepted you and offered you good funding. (The tricky part of course is if you have to choose between less prestige and more funding, or more prestige and less funding; statistically people seem to end up better off if they value funding over prestige---simply getting into a top school says a lot about what you're capable of, even if you never go there.)
But from society's perspective, there is a much bigger question: What is the purpose of higher education?
If our goal is to take the very smartest people---the top 10% or even 1%---and give them the resources they need to become a maximally efficient technocratic elite of scientists, doctors, engineers, and lawyers, then expensive private schools are accomplishing that goal well, because they provide top-quality education and price isn't all that important.
But if our goal is to provide equal access and opportunity to the entire population to provide upward mobility, public schools are far more effective at that goal, because their lower prices and more financial aid provide more opportunities for students from disadvantaged backgrounds to go to college.
Different people may disagree on which of those goals is most important, and how to trade off between them.
In practice, almost all schools, regardless of their official type, are some hybrid of private and public. The University of Michigan retains a large endowment of private donations, and Harvard receives a good deal of public funding; it's just that Michigan has a lot more public funding and Harvard has a much bigger private endowment, so we consider Michigan a public university and Harvard a private university. This hybrid model may reflect our disagreement and ambivalence about the two purposes of education; we're trying to do both at once.