there is such a wide avriety of public and oprivate schools in the US that quaklity cannot be measured by that feature alone. The very finest schools in terms of quality do tend to be private schools, i.e. Harvard, Stanford, Princeton, Yale, but some of the top schools are public ones, such as University of Michigan, Berkeley, UCLA.
One difference is usually size, as private schools tend to be smaller, but the classes may be larger. E.g. I took a Spivak style honors calculus at Harvard in a class of 135, whereas at UGA this class has probably a dozen students.
But even within the same school there are great variations in class size and quality of instruction. The class I took was large so that one outstanding professor could lecture to all the students. Other classes in math at the same school were small and taught by grad assistants.
One other common difference is cost. Since they are not subsidized by the legislature, private schools are almost always more expensive tha public schools. And the price has little relation to the quality. Mediocre private schools cost almost as much as the outstanding world famous ones.
The student bodies may also differ, since many public schools are more open to average students from the local area, hence may contain many more weak students, but some private schools enroll weak students too.
A public school is more likely to have a large student body, more diverse with respect to academic ability, with more low achieving students, and more group activities, like big football teams.
But an elite private school may be more diverse in terms of international representation, and talent, a place where one is more likely to meet a variety of new people. But there are also small insular private schools, particularly religious based ones, where there will be esentially no diversity at all. The primary universal feature is that private schools cost more.
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