While it's true that there are significant differences in resources available at public v. private schools, I am not entirely certain this trickles down to individual student-level experience in the manner suggested at the link. Take, for a non-random example, my own experience in political science at the University of Michigan: Introduction to American Politics had 500 people, and was taught by a GSI in section with, as best I recall, 25 or so people. Introduction to Political Theory had 400, and a similar discussion-section arrangement. Most of my upper-level classes were around 40 students--40 meant no discussion sections, just the professor.
In spite of this, I had exactly as much attention from the professors in these classes as I wanted--which in a few instances was significant contact over multiple years. In fact, I probably had comparable interactions to those students who went to private schools (at least those who never wrote a thesis). The reason is that the culture of a public school is radically different than that of a private school: students at public schools tend to think of class as a place you go to absorb knowledge and then leave, and the bigger the class, the greater the expectation of no interaction. This meant that, even in those 40-person classes with high student interest, I was competing with--at most--ten other people for the professor's time, and usually fewer than that; at least in my experience, that's equivalent to the more time-intensive approach of private schools.
The lack of resources still matters: it makes a difference in the ability to attract and retain top faculty, and it makes a difference in how the average student is served (I was not the average student), who is unlikely to take advantage of opportunities even if they have some desire to do so because the culture of the school disinclines them to do so--and if this is true at Michigan, which is certainly academically-oriented, I can only imagine it to be more true at other public schools with fewer resources.