Ok so I set up two factor verification and now needed to set an app specific password for Fantastical 2. Now I have it on two computers and it prompted me to make a password for each. So I generated one for each and both work fine. Now I'm still confused with this. Do I need to save those passwords or write them down? Will I need them again? This whole two-step/two factor thing is so confusing and their literature on it doesn't exactly explain it well.
I have Fantastical 2 and did the app specific password just like you. I did not write it down since it is save in OS X Keychain, which I backup. If you open Keychain app and search for Fantastical you will see where the application password is save. Just double click the entry then enter your login password and you will be able to see the app specific password there.
Unless want to reinstall on another Mac or something, you won't need to reenter this anywhere.
If for some reason you were to lose this password or forget it, you could just login under your AppleID and in the mange passwords section you could disable this password and issue a new one.