I have an old 20GB Xbox HDD from my original white Xbox. I also have an xbox transfer cable. Would it be possible to use this transfer cable to use the hdd as an external drive for my PC? I plugged it in and it downloaded the device driver software but the hdd does not appear in "computer".

It won't appear in "Computer" because it's not formatted in a format Windows can read. If the drive is good, It'll appear in diskmgmt.msc / disk management. You will then be able to format it in a Windows-friendly format and use it as an external.

Of course a 20 GB external can be trounced by a $20 USB stick...

Hell, they're cheaper than that. My 32GB was $10