Generally, school prestige is a holdover from way back then. A little bit of private school elitism. A little bit of education not being as standardized. Harvard, Yale, etc cemented their reputation as Hugh class institutions long before college became widely available.

At a practical level, the math you learn at your state university is the same as at Harvard. The value of a high class institution is not necessarily in the actual education.

As prestigious universities, their degrees carry weight and they also have lots of connections. As the adage goes, it's not just what you know, it's who you know. So prestigious universities open more doors simply because they have better connections.

In terms of education, it is true that prestigious universities intentionally try to draw some of the top scholars who are doing cutting edge research. However, most of the time they're doing research and even if they do teach it is rarely undergrad classes. Top scholars usually only matter for people doing postgraduate study as they will join in their research.

So while there are some academic advantages, the real difference between prestigious universities and Podunk State down the street is social.