How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?
Not too much, I don't belive that Twin PRime suggest much for Riemann (unless its some statistical stuff or basic guarantees).
The Twin Prime Conjecture is really part of a series of other conjectures which looks follows.
"While the Riemann Hypothesis is decisive in determining the distribution of primes, it seems to be of little help with regard to twin primes."
"The conjecture that the distribution of twin primes satisfies a Riemann Hypothesis type error term is well supported empirically, but I think this might be a problem that survives the current millennium."
However, the first part of your question is more general, in that you ask about how proving one of these conjectures/hypotheses would affect the other. That is a more nebulous question, because it's hard to predict what sort of machinery will ultimately be developed to resolve these questions.
If this answer sounds like a bit of a downer to you, perhaps there is a silver lining: given two large conjectures that don't really have much bearing on one another, it would/will be interesting to see how the resolution of both could be applied elsewhere. If they were intimately connected, solving one might knock the other one off; with their relation tenuous at best, it ought to take some wonderful mathematics to dispose of both. It remains to be seen what can be done with the machinery resulting from each - I just hope it is seen in our lifetimes!