The major differentiator between private and public schools, IMO, is that privates, as businesses, have inherent flexibility that can't be matched by larger scale, bureaucratic organizations. That allows them to be (at least theoretically) more responsive to the client. The lack of a unionized teaching workforce gives the parents greater bargaining power in the school, which results in an experience that can (absent terrible management) be better tailored to the students' needs.
Well-run private schools adopt an efficiency wage model, whereby teachers are paid a premium in gross compensation over their public sector counterparts. This aligns the incentives of the teaching staff and the administrators, creates a disincentive for deficient teaching and mentorship, and mitigates agency problems present in public schools.
So, really, it's a situation that uses economic incentives to align the interests of parents and teachers around a common goal. With a double-sided positive self-selection bias, teachers and students are positioned more collaboratively in private schools simply by dint of a common purpose.
