The first response to this question is about money. Public universities are funded by state governments while private universities are not rather, their monies come from private contributions causing their tuition to be somewhat higher. A second difference would be size. Public universities tend to have a much higher student population which makes sense because the private universities with higher tuition rates, have much smaller populations. Along these lines, you can also see how class size would be impacted by student population. Once more, the larger the overall population, the larger the class size tends to be. Majors are another area where public and private institutions differ. The larger the school (public), the more majors, the smaller the school, less offerings in terms of majors. Then there are other areas to consider such as the differences in the social aspect of each type of school. Students attending Penn St. for example, can take part in large stadium football games while those at a private university, may not have this type of large sporting event experience. There tends to be greater research at public universities due to their size as well.
Keep in mind that a quality education can be obtained by students whether attending a large, public university or a small, private institution. As with many aspects of life, it is what one puts into their experience which, will determine how a student benefits for their time in college.