First, please only post your issue once. Duplication causes confusion. Second, compare setups between the two PCs. It seems that one may be POP, downloading and removing from server, and then the other PC doesn't see the messages. Ideally, both PCs should be IMAP if you want both to show the same.

Thanks David, Guess, it was my screw-up. My main computer had SMTP Settings, Laptop had POP settings. Is there a way to get it back on Server from this Laptop. I meanwhile changed the Laptop settings to SMTP. But Thunderbird doesn't communicate with it any more. It will not send or receive mails. I don't want to loose these emails. If I re-install user again, will it keep the emails?

You could create IMAP account on laptop. Then, you could copy the messages in the POP account inbox to the inbox for the IMAP account. You CANNOT change a POP account to IMAP. Not a problem if the messages are still on the laptop. Keep the POP message folders alongside the IMAP account and remove it after all has been set straight.