What’s the difference between “New Threads” and “New Messages” when watching a forum?
Messages presumably refers to direct (private) messages.
In the context of a forum though … maybe it refers to any new post. Give it a try and let us know. 
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This whole page is a “thread”. A “thread” is a conversation which is listed in the forum.
A “message” is a single contribution by one person to a thread. They’re often referred to as “posts”, which is a generic term for them, and “message” is the word used by this specific forum software.
So I think what that’s asking is whether you want it to watch for new conversations starting, or for new conversations and any replies to them. I’d not be 100% confident until I’d tried it, though.
I did some tests.
“Message” indeed means a “single contribution/post” by one person to a thread.
However, either e-mail alerts from a new thread or a new message contains the note in the footer.
And I confirm the e-mail behavior is just like it said, neither a new thread alert nor a new message (from a old thread created before you watch the form) will get the following messages of its thread, unless you click the view this thread button to open the web page.
It seems impossible to make the forum e-mail alert behaves like a traditional e-mail listing. Though it is still acceptable behavior for watching a forum - you won’t get the following alerts of a thread that you’re not engaged in yet.
What I don’t like is the same behavior pattern for a watched thread. As the thread owner/starter, I get this note
I have watched (subscribed) to a specific thread, why do I have to click into the forum web page every single time to convince Unity Forum that I’m still interested in the thread?