It's the same story as with many other deep and important math problems: Practically, not much would change. Many papers already assume RH - we would just have

It's the same story as with many other deep and important math problems: Practically, not much would change. Many papers already assume RH - we would just have confirmation that they are reality then. What would change is that we would likely have novel and powerful methods that were used in the proof. This is what most of the buzz is about. For example, the proof of Fermats Last Theorem introduced novel connections between sub fields of mathematics.