How each is funded, and what that changes

A public institution is substantially funded by the state and charges fees to cover the rest. A private one is funded by its fees, and sometimes by an endowment or an owner. That single difference explains most of what follows from it.

It means a public institution's incentives are shaped by public funding rules and regulation, and a private one's are shaped by enrolment. Neither is automatically better for a student: public funding can bring rigid structures and very large classes, and fee dependence can bring responsiveness or it can bring pressure to admit. What matters is knowing which pressure applies to the place in front of you.