iCloud email password

Where are you trying to use your iCloud e-mail address? If in a 3rd party app/application, you need an app specific password.

Apple ID -Two-factor authentication

Two Factor Authentification availability

Apple ID - Using app-specific passwords

Apparently my iCloud email password is different than my Apple ID password.

This would suggest you have 2 Apple IDs. You manage their passwords separately, using the same steps.