I have several gmail accounts linked through IMAP to both Thunderbird and my iPod touch.
By using IMAP the messages are kept on the server, and by moving them about in my email client the changes are mirrored on the server. POP downloads the messages from the server to my client, and there's no feedback to the server.
If I apply several labels through the gmail interface in the browser, then the message will be 'duplicated' in each of those corresponding folders in Thunderbird. (With IMAP).
If I want to apply serveral labels usnig Thunderbird, then I need to hold down the shift button (I think ?) whilst dragging the message to another folder. On gmail this applies two labels (or more) to the message, whilst not duplicating the message. If you do this action with a POP account then your message gets duplicated - but only locally, changes made on the client are not reflected in a POP account on the server.
Hope this helps.