What advantages do law students receive by attending a law school like Harvard, other than name recognition?
A prestigious law school has benefits. But, answering the question depends on how a law student defines “better education.”
Will the prestigious law school give law students a better knowledge of black letter law? Absolutely not. Everyone, even at prestigious law schools learn black letter law during bar prep courses prior to the bar exam.
What law students will learn in school is the principle of how to think like a lawyer and a small amount of basic law in fundamental subjects.
At a prestigious school, law students learn these things from renowned law professors. This might amount to a slightly better education but practically speaking there is likely no difference between a prestigious law school (say top 3) and any other law school in the top 20 or even top 50. All of these schools have some high quality law school faculty.
The real advantage that comes from attending a law school like Harvard is the connections. Law students graduating from a top law school will have the best network in the world to draw on for internships, clerkships, and mentorship. Search any court, law firm, or fortune 100 company in the United States and you can find someone that went to Harvard law. Going to Harvard law substantially increases your chances at connection with your alumni and possibly getting an opportunity as well.
Obviously, this is not how it works in every situation. There are plenty of Harvard graduates turned down when they thought a fellow Harvard alumni working at their desired placement would help. And, many people rise to all sorts of positions without the benefit of a Harvard Law network.
Conclusion
The short answer is that no. Generally, law students will not receive a better education by attending a prestigious law school over another top law school. If you attend a really low ranked low school, that is a different story. The main advantage of prestigious law schools is in the alumni network law graduates become a part of.