Setting and changing email passwords
Thunderbird doesn't provide a way for a user to enter the POP/IMAP/SMTP servers password when configuring a account. This frequently confuses new users who expect to be able to do that because they did that with their previous email client. Thunderbird will prompt for a password the first time it needs one. When it does that you can check the checkbox to save the password using the password manager if you want Thunderbird to remember the password.
When you change a password using Thunderbird you're merely changing Thunderbirds copy of the password. It has no effect on the real password. You typically have to use a browser and login to webmail to change the password that the mail server expects.
The only way to change a password saved by the password manager is to delete it, get prompted for the new password the next time it needs it, and then tell it again to save that password. You can delete the password using Tools -> Options -> Privacy -> Passwords -> View Saved Passwords by selecting the password and then pressing the Remove button.