Why won't my computer see my Galaxy S III as an external drive when plugged in via USB?

Leo says that's by design. What he needs to do is put the phone into "disk mode," or "MTP mode" so that the computer will see it as a disk to mount. This should be in the settings. If it doesn't mount, try a different cable. If that doesn't do it, then there's something wrong and it could be hardware or software at this point. A driver may not be loading that the phone needs to broadcast that it's there. Resetting the phone will reload that. If that doesn't work, then take the phone back.