Use twin primes for encryption. Twin primes are employed in algorithms developed for communication security and for the transmission and compression of information --- in cell phones, for example. Computers might use twin primes to protect money transfers between banks.
Advance the cause of mathematics. When you start looking for twin primes, preferably with a large computer, you keep finding them. No one has discovered whether or not there is an infinite number of twin primes --- it's one of the unsolved problems of mathematics. The proof --- one way or another --- is out there waiting.
Revel at the complexity in simplicity. While twin primes are the ultimate simple things --- they can be divided only by themselves and one --- they are deeply puzzling. Norwegian mathematician Viggo Brun proved in 1919 that the sum of the reciprocals of successive twin primes, converges to a value now called Brun's Constant, indicating that twin primes, while relatively scarce, might be infinite in number.