Why the search for ever larger primes?

There was a significant real-life effect to the effort to discover primes: it was how the infamous Pentium bug was discovered.

Professor Thomas Nicely, then a professor of mathematics at Lynchburg College, had written code to enumerate primes, twin primes, prime triplets, and prime quadruplets. Nicely noticed some inconsistencies in the calculations on June 13, 1994 shortly after adding a Pentium system to his group of computers, but was unable to eliminate other factors (such as programming errors, motherboard chipsets, etc.) until October 19, 1994. On October 24, 1994 he reported the issue to Intel. According to Nicely, his contact person at Intel would later admit that Intel had been aware of the problem since May 1994, when the flaw had been discovered when testing the FPU for its new Intel P6 core, first used in the Pentium Pro.

Well the M in GIMPS stands for Mersenne, and it hasn't been proven that there are infinitely many Mersenne primes. But it's widely believed to be true--in fact there is a conjectural estimate of their frequency. I think the search for Mersenne primes is mostly a "because it is there" thing, but it could provide numerical evidence for or against this conjecture.

GIMPS is probably more interesting as an experiment in massively distributed computation.

The question is answered to some extent in The Prime Pages FAQ.