Is the email account set up as an IMAP or POP3 account?
When you say that incoming works OK is this on T'bird or when using webmail?
As you may know, when setting up T'bird the automatic default when entering the account password is to use the same for Incoming and Outgoing servers.
With that in mind it appears odd that only the outgoing password appears to be the problem.
Before doing anything ensure that you are using the most up to date version of T'bird and if an POP3 account ensure that you do a back up of the downloaded emails and or check that they are still on the BT servers by logging onto webmail to ensure they are still in the Inbox and can be re-downloaded.
If an IMAP account the emails showing should still be the same as those on the BT servers.
I would then remove the email account from T'bird or delete and reinstall T'bird and then set it up manually and enter all the details rather than letting it be set up with the automatic wizard. This includes entering the incoming ad outgoing passwords manually rather than using the "use the same as " option.
The settings you need are as per the following link.
