Finite fields, essential to error-correcting codes, are based on properties of primes. This forms the basics of efficient data transfer, whether Internet, CDs or communication with spacecrafts. The book "Number Theory in Science and Communication" mentions that finite fields were used in verification of the fourth prediction of general relativity. See that book for more examples.

Here's one from http://mathoverflow.net/questions/2556: There is a gamma ray telescope design using mod p quadratic residues to construct a mask. Gamma rays cannot be focused, so this design uses a redundant array of detectors separated from the mask to reconstruct directional information.

Cicidas have a cyclic life spanning a prime number of years. It is supposed that this is because a predator has harder job aligning with the cycle. http://en.wikipedia.org/wiki/Predator_satiation

I believe that the Möbius function is important in advanced physics: http://en.wikipedia.org/wiki/Mobius_function#Physics

There's a legend that generals used the Chinese remainer theorem to count soldiers.

Elementary number theory was a motivating factor and foundation for development of the whole mathematics - algebra, analysis, even logic. Did you know that proving Godel's incompleteness theorem requires using primes? http://mathoverflow.net/questions/19857/has-decidability-got... Transcendence of pi also needs primes. http://mathoverflow.net/questions/21367/proof-that-pi-is-tra.... I did not realize that earlier. Usage of primes is often invisible. Even if there was no cryptography or ECCs, it would be silly to demand practical applications from this subject - it is a foundation for almost everything in mathematics, sometimes rather concealed.