What does proving the Riemann Hypothesis accomplish?
Proving the Riemann Hypothesis will get you tenure, pretty much anywhere you want it.
The Millennium problems are not necessarily problems whose solution will lead to curing cancer. These are problems in mathematics and were chosen for their importance in mathematics rather for their potential in applications.
There are plenty of important open problems in mathematics, and the Clay Institute had to narrow it down to seven. Whatever the reasons may be, it is clear such a short list is incomplete and does not claim to be a comprehensive list of the most important problems to solve. However, each of the problems solved is extremely central, important, interesting, and hard. Some of these problems have direct consequences, for instance the Riemann hypothesis. There are many (many many) theorems in number theory that go like "if the Riemann hypothesis is true, then blah blah", so knowing it is true will immediately validate the consequences in these theorems as true.
In contrast, a solution to some of the other Millennium problems is (highly likely) not going to lead to anything dramatic. For instance, the $P$ vs. $NP$ problem. I personally doubt it is probable that $P=NP$. The reason it's an important question is not because we don't (philosophically) already know the answer, but rather that we don't have a bloody clue how to prove it. It means that there are fundamental issues in computability (which is a hell of an important subject these days) that we just don't understand. Solving $P \ne NP$ will be important not for the result but for the techniques that will be used. (Of course, in the unlikely event that $P=NP$, enormous consequences will follow. But that is about as likely as it is that the Hitchhiker's Guide to the Galaxy is based on true events.)
The Poincaré conjecture is an extremely basic problem about three-dimensional space. I think three-dimensional space is very important, so if we can't answer a very fundamental question about it, then we don't understand it well. I'm not an expert on Perelman's solution, nor the field to which it belongs, so I can't tell what consequences his techniques have for better understanding three-dimensional space, but I'm sure there are.
Explaining the true mathematics behind the Riemann Hypothesis requires more text that I'm allotted (took most of my undergraduate degree in mathematics to even touch the surface; required all of graduate school to fully appreciate the beauty).
In very simple terms, the Riemann Hypothesis is mostly about the distribution of prime numbers. The idea is that mathematicians have some very good approximations (emphasis on approximate) for the density of the primes (so you give me an integer, and I can use these approximate functions to tell you roughly how many primes are between 0 [really 2] and that integer). The reason we use these approximations is that no [known] function exists that efficiently and perfectly computes the number of primes less than a given integer (we're talking numbers with literally millions of zeros). Since we can't determine the exact values (again, I'm simplifying a lot of this) the problem mathematicians want to know is exactly HOW good are these approximations.
This is where the Riemann Hypothesis comes in to play. For well over a century, mathematicians have known that a special form of the polylogarithm function (again, more fun math if you're bored) is a really great approximation for the prime counting function (and it's way easier to compute). The Riemann Hypothesis, if true, would guarantee a far greater bound on the difference between this approximation and the real value. In other words, the importance of the Riemann Hypothesis is that it tells us a lot about how chaotic the primes numbers really are. That's an incredibly high-level explanation and the Riemann Hypothesis deals with literally hundreds of other concepts, but the main point is understanding the distribution of the primes.