The difference is that all known attacks on AES require in the neighborhood of 2^length attempts to succeed; that is, there's no better method known than simply trying different keys by brute force.

It follows, then, that a 256 bit key is 2¹²⁸ times as hard to crack as a 128 bit key.

Of course, computing each encrypted block with 256 bit keys is harder too.

The choice of key length should be based on risk: the damage that will be done if an attack is successful.

The difference is that all known attacks on AES [but see comments] require in the neighborhood of 2^length attempts to succeed; that is, there's no better method known than simply trying different keys by brute force.

It follows, then, that a 256 bit key is 2¹²⁸ times as hard to crack as a 128 bit key.

Of course, computing each encrypted block with 256 bit keys is harder too.

The choice of key length should be based on risk: the damage that will be done if an attack is successful.