Where the difference actually shows up

Not usually in teaching quality, which varies far more within each category than between them. It shows up in class size, in how much teaching is done by permanent staff, in facilities, in how quickly a course can be changed, and in what happens when something goes wrong.

The variable that actually predicts your outcome is the department, not the sector. A strong department at a mid-ranked public university beats a weak one at an expensive private institution, and the reverse holds just as often. Judge at department level, because that is where the teaching happens.