From what I understand, the main premise of the twin prime conjecture is "Are there an infinite number of twin primes?" And twin primes are prime numbers that are separated by two. Examples include: $(3,5), (5,7), (11,13), (17,19)... (793517,793519), (793787,793789), (793841,793843)... (2924351,2924353), (2924567,2924569), (2924921,2924923)... (7120187,7120189), (7120277,7120279)... (12382691,12382693), (12382691,12382693)... (16148159,16148161)... (17355509,17355511)... (18409199,18409201)$, etc.
If I have something wrong, please tell me. If I have it correct, please explain to me why this matters. What I mean by why it matters, is what effect will it have on the real world. Usually when I hear of the practicality of prime numbers, it is in reference to cryptography. So, if there are an infinite number of twin primes, does this mean good for white hats, bad for black hats? And what if there are not an infinite number of primes and we learn them all. What implications will that have in the real world. Does the importance of the twin prime conjecture go beyond cryptography? If so, please explain. Thank you.
A solution of the twin prime conjecture most likely will show completely new ideas and techniques in the area of analytic number theory. The result itself will not really matter for applications, but the methods might be very helpful for later applications.
As for real-life applications of prime numbers in general, there has been said more than enough on this site, compare for example this post (and others):
Real-world applications of prime numbers?