The issue here is how complicated is each statement, when formulated as a claim about the natural numbers (the Riemann hypothesis can be made into such statement).
For the purpose of this discussion we work in the natural numbers, with $+,\cdot$ and the successor function, and Peano Axioms will be our base theory; so by "true" and "false" we mean in the natural numbers and "provable" means from Peano Arithmetic.
We will say that a statement is "simple" if in order to verify it, you absolutely know that you do not have to check all the natural numbers. (The technical term here is "bounded" or "$\Delta_0$".)
For example, "There is a prime number small than $x$" is a simple statement, since verifying whether or not some $n$ is prime requires only checking its divisibility by numbers less than $n$. So we only need to check numbers below $x$ in order to verify this.
On the other hand, "There is a Fermat prime larger than $x$" is not a simple statement, since possibly this is false but only checking all the numbers above $x$ will tell us the truth of this statement.
The trick is that a simple statement is true if and only if it is provable. This requires some work, but it is not incredibly difficult to show. Alas, most interesting questions cannot be formulated as simple statements. Luckily for us, this "provable if and only if true" can be pushed a bit more.
We say that a statement is "relatively simple" if it has the form "There exists some value for $x$ such that plugging it in will turn the statement simple". (The technical term is $\Sigma_1$ statement.)
Looking back, the statement about the existence of a Fermat prime above $x$ is such statement. Because if $n>x$ is a Fermat prime, then the statement "$n$ is larger than $x$ and a Fermat prime" is now simple.
Using a neat little trick we can show that a relatively simple statement is also true if and only if it is provable.
And now comes the nice part. The Riemann hypothesis can be formulated as the negation of a relatively simple statement. So if the Riemann hypothesis was false, its negation was provable, so Riemann hypothesis would be refutable. This means that if you cannot disprove the Riemann hypothesis, it has to be true. The same can also be said on the Goldbach conjecture.
So both of these might turn to be independent, in the sense that they cannot be proved from Peano Arithmetic, but if you show that they are at least consistent, then you immediately obtain that they are true. And this would give us a proof of these statements from a stronger theory (e.g. set theory).
You could also ask the same about the twin prime conjecture. But this conjecture is no longer a relatively simple statement nor the negation of one. So the above does not hold for, and it is feasible that the conjecture is consistent, but false in the natural numbers.
Best Answer
The distribution of primes is not going to change. No matter what we discover about the Riemann hypothesis or any other area of math, the distribution of primes will not change.
The Riemann hypothesis implies a bound on the error term in the prime number theorem. Specifically, it implies that $\pi(x)=\frac x{\log x}+O(\sqrt x\log x)$. If the Riemann hypothesis is shown not to be true, then we will not know that this result is true. (I believe, though I may be wrong, that the result is implied by but not equivalent to the RH; correct me if I'm wrong.)
Now, any theoretical proof that the RH is false (or true) would almost certainly involve theory which would cast further light on the distribution of the primes in some regard which might be more valuable than the disproof (or proof) of the RH itself. A discovery of a zero not on the critical line would of course be less helpful in this regard.
In either case, it is unlikely that the RH has any direct connection to the twin prime conjecture, since twin primes occur with frequency at most $\frac 1{\log x}$ in the primes (Brun's theorem), so a bound on the error term in the prime number theorem is probably too specific a result to have much effect. The twin prime conjecture is more closely related to the occurrence of primes in polynomials, and so to conjectures such as Schinzel's Hypothesis H or the Bateman-Horn conjecture (each of which imply the twin prime conjecture) or Bunyakovsky's conjecture (a weaker version of the above two which does not).