Is the account on the home computer POP (Server Type in Account Settings), while it's IMAP on the laptop? It isn't related to the TB or Windows version. Most users who access accounts on multiple devices set them up as IMAP to sync all with mail stored on the mail server.

Yes, the home computer is POP and the laptop is IMAP, but the issue is that the Server Settings screen on the laptop is missing the option LEAVE MESSAGES ON SERVER just below FETCH HEADERS ONLY.

However, I just did a test and (1) incoming messages that came to the laptop while the home computer was turned off also come to the laptop when it is turned on, and (2) deleting them from the laptop does not result in deleting them from the home computer -- so this set-up must permit messages to remain on the server. Wish there was a tech note stating this.

So, it seems that this is a non-problem.

My big concern is that I teach 50-plus students at two universities and accumulate many messages a day -- so when I am away from home I respond with a bcc to my e-mail address. When I come home, the hundreds of e-mails are still on the server, together with my response -- and they download to my home computer when I start it up again. But when I installed Thunderbird on my new computer, there was no option to leave messages on the server. Coming home to having to copy hundreds of messages to my home computer is not a requirement I want to have. On all previous laptops I had the option to leave messages on the server, but the options (for the SAME version) of Thunderbird are different on this laptop.

The option 'leave messages on server' isn't offered for IMAP accounts, because the main feature of IMAP is that mail is stored on the server, and is synced with whatever mail client is connecting with an IMAP account. If you have a POP setup on one device and IMAP on another, you will soon find conflicts and inconveniences.

https://support.mozilla.org/en-US/kb/difference-between-imap-and-pop3