I've been instructed to backup my wallet's wallet.dat file.
However, in my wallet software (Multibit), I don't see a backup option other than "Export Private Keys", which saves a .key file.
What is the difference between copying the wallet.dat file and creating a ".key" file using the Export option?
Well both contains private keys.
Multibit is saving as a .key file.
Here some official instructions how to import private keys from a .key file.
https://multibit.org/help_importASingleKey.html