Back in the 20th century there was a goal to ensure maths was consistent, notably by David Hilbert, and that everything could be proven from a finite set of axi

Back in the 20th century there was a goal to ensure maths was consistent, notably by David Hilbert, and that everything could be proven from a finite set of axioms, and he wanted to ensure everything was built on a logical foundation and was rigorous. In practice it may not change much, but more philosophically speaking, it's dangerous to take something as granted without a proof. If we then have many ideas building off this unproven idea, and it's later shown to be false, a lot of those theorems would be based off a false premise. This doesn't necessarily mean they the proposition is incorrect, but it would mean that they are unproven. It's more to satisfy the rigour in maths, and also pose as a challenge to mathematicians. In the journey to proving RH a lot of new ideas could be found that could prove useful, too.