If I understand the problem correctly, whichever email client (Outlook Express or ThunderBird) first opens the email removes it from the server, so it is no longer available. In Thunderbird, click the Tools, then Account Settings. Then click Server Settings. Put a check in the box "Leave message on server." You probably should also check the "For at most" box and fill in 7, 10 or whatever number of days is reasonable to ensure you get the email with any client and on any computer you use. Also check "Automatically download new messages."
You can do the same thing in Outlook Express by clicking Tools>Accounts>Properties>Advanced.
I agree that ThunderBird is much more preferable than Outlook Express and once you get ThunderBird tweaked properly and used to it, I would dump Outlook Express.