I currently have access to one of 2 computers at any given time, without too much choice as to which one I get. If I want to play with the same character on both computers do I simply need to copy the folder "<character name>" from the most recently used computer to the current one before logging in each time, or is the settings and config system more subtle than that?
If I don't copy anything across, what do I lose out on?
You'll only know the names of people you've been introduced to on the computer your using, and you'll only be able to see parts of the map you've explored on the computer you're using.
Other than that, I don't believe there are any technical issues.
You can do what you described, yes. But you can also try to use an USB-stick; just set the drive of the USB stick (has to be the same on both computers referring to Nop) as your Illarion folder.