What is the significance of studying this problem? I mean, say they prove the twin primes conjecture. What does that mean? Do we benefit like we would if the travelling salesman problem was solved?
Significance of a problem shouldn't be considered from the same point of view as pricing an asset, by face value.
Open problems concerning prime numbers can be considered as large floodgates, waiting to be opened. Their applicability is likely to be infinite. Pick up any subject, try to quantify some behavior, sooner or later, primes will make an appearance. In fact, you should really do this; read the first few chapters of an elementary number theory book, then try to represent your area of expertise in a way that involves primes.
Perhaps in a few decades, or centuries, by tracing a certain technological improvement back in time, we'll be able to place a $-value this work. You can already do this now, for other (probably all, if you can be bothered) areas of (previously abstruse) mathematics, for example Group Theory -> Spectroscopy -> Biomedical Spectroscopy, or Algebraic Topology -> Improvements in semiconductors.
This is a different but related problem: instead of trying to prove that there are arbitrarily large twin primes, they are trying to see how the maximum gap between primes grows with their size. I know that doesn't answer the applicability question, but I think the answer to that is "who knows?" Pure mathematicians tend to study these things for their own sake and then someone may or may not figure out something useful to do with it.
Think of it more like fundamental research. Maybe nobody could make immediate productive use of the discovery of (for example) radioactivity in the late 1800s, but it has a wide variety of uses today from energy production to medical imaging.
It's unlikely that proof of the twin prime conjecture would prove that fundamental, but the results are hard to predict in advance.
