That's expected for apps connected with Exchange, ActiveSync or MAPI: once two-factor authentication is on, those stop accepting your main password straight away. To fix it:

1. Create an app password for that app, with the right box ticked.
2. In the app's account settings, replace your old password with the app password.

Apps connected with IMAP, POP3, SMTP or CalDAV aren't affected by turning on two-factor authentication. If one of those has stopped working, check the password it has saved. You'll often see this as "535 Incorrect authentication data" (see Error 535).