Actually the screenshots you've shown here show the POP3 settings in Outlook and the SMTP settings in Thunderbird. In the top one the SMTP settings are correct I guess (mail.btinternet.com) but the logon information there is for the POP3 box, not the SMTP connection. This is no problem as the SMTP server normally doesn't require authentication.
On the bottom dialog I also suggest unticking the box at the top with 'outgoing server requires authentication'. As long as the STMP server (same as above) is entered correctly on the 'General' tab then you should be ok.
One other thing to check is whether your firewall allows Thunderbird access to the internet, perhaps the outgoing port is blocked by accident.
Edit: Disregard, I misidentified the bottom dialog. It is the 'more settings' dialog from Outlook I guess. You could try what I've suggested above in Outlook and see whether the server accepts mail without the logon info.
Another (free) e-mail client is Pegasus, available here:
http://www.pmail.com. Although with a bit of tinkering I'm sure Thunderbird will do whatever you want.