Even for AES128 we don't use the key directly. Each round gets its own key, which is not in the supplied bitsprocess called key expansion. Those round keys are used to transmute a matrix. The original key is never applied directly.
AES256 adds more rounds. Each round stays the same, only the key derivation for them changes slightly. This allows for less dependant round keys, which hopefully adds to the cipher strength. However, there is some deficiency in the key expansion algorithm which makes the 192 and 256-bit variants more susceptible to related key attacks.