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AddRoundKey. That step takes 16 bytes from the expanded key schedule, and exclusive-or's ("adds" in $GF(256)$ terminology) it to the intermediate block state.
The keyword you are looking for is related-key attack, where exactly this kind of different input keys (without the restriction to 1 byte) is used to break the cipher. (Wiki) Related crypto-SE topic: Related-key attacks on AES A related-key attack on AES has been analyzed by Biryukov and Khovratovich Also worth a look: Security section of Wikpedia for a ...
How the cipher key is scheduled if it is less than 128 bit (for AES 128), or less than 192 bit (for AES 192) or less than 256 (for AES 256)? That is undefined; the AES specification does not address that possibility. The AES 128 algorithm assumes that you give it 128 bits of key (and tells you exactly what to do with that key); it says nothing about ...
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