[crizCraig]

The two main consequences that would follow are (from Wikipedia):

"A proof that showed that P ≠ NP, while lacking the practical computational benefits of a proof that P = NP, would also represent a very significant advance in computational complexity theory and provide guidance for future research. It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place."

"Cryptography, for example, relies on certain problems being difficult. A constructive and efficient solution to the NP-complete problem 3-SAT would break many existing cryptosystems such as Public-key cryptography, used for economic transactions over the internet, and Triple DES, used for transactions between banks. These would need to be modified or replaced."

So basically we can focus on finding good approximations to NP problems and feel safe that this proof won't immediately jeopardize all of our bank accounts.

[NateLawson]

The above is false (par for Wikipedia). While RSA is based on the difficulty of factoring, DES (and 3DES) are not. The DES cipher is based on substitution/permutation and not number theory.

Also, factoring is known to be subexponential (e.g., GNFS). While there is no known polynomial time factoring algorithm, this may give some evidence that the integer factorization problem might be in P. Factoring is known to not be NP-complete and we hope it is not in P. While a proof that P=NP would be disastrous for RSA, a proof that P!=NP does not mean factorization is guaranteed to be safe against future advances in algorithms.

In other words, proof that P!=NP would be an amazing result but someone could still improve factoring algorithms.