By what main criteria does the mathematical community judge whether an open problem is important? By what main criteria does the mathematical community judge whether an open problem is important among many others open problems in mathematics? For example if we say that the Riemann Hypothesis is an important problem because it is related to the prime distributions, are all open problems related to prime distributions important?
An open problem is important if its resolution affects many, diverse areas (indeed, the farther flung the impact, the more significant the problem).
In the case of the Riemann hypothesis, there are many theorems that are conditional on it (see Consequences of Riemann hypothesis), though they are largely in the broader realm of number theory.
Other problems, like the P vs NP problem could have consequences beyond automata theory that include cryptography (and, by extension, our entire
e-commerce industry), computational biology, manufacturing and optimization, etc.