Actually, TB is not a just mail checker, it's a mail client.
A mail checker is a small tool that can check and notify you about new
messages for you. So it is simply looking for new messages on the
according server and, if any, notifies you about them.
Therefore you may also call these tools mail notifier.
A mail client has way more features than that. You can download your
messages and you can create or edit messages and send them, etc...
Therefore, a mail client is much bigger.
So, usually you would rather want to start a mail checker than a mail
client, because it requires less ressources.
Unfortunately, on Linux (in my case Linux Mint) the available mail
checkers are so bad to set up, that you better start TB right at boot up
and put not only in the task bar, but in the tray, because you better
have a tiny icon in the tray than a big button of a minimized
application in the taskbar.
By doing so, you get an icon looking like an envelope and when new
messages are coming in, that icon shows the number of the new messages.
The big advantage of the add-on alternative I'm looking for is, that it
can minimize TB automatically (as I said, at boot up) and do its job
(most of the time) inconspicuously in the background.