For a motivated learner with access to good materials, schools provide two important things besides that very important piece of paper:

1. contacts - these come in the form of peers who are interested in the same things and in the form of experts in their fields of study. Talking to these people and developing relationships will help you learn faster, and teach you how to have professional collegial relationships. These people can open doors for you long after graduation.

2. facilities - ever want to play with an electron microscope or work with dangerous chemicals safely? Different schools have different facilities available for students in different fields. If you want to study nuclear physics, you might want to go to a school with a research reactor; it's not a good idea to build your own.