Man (in the link, see meaning 2.1; in singular it means human beings in general; the human race) is an old English word which was used to refer to all people in general. That's where the word "mankind" comes from: the kind of all people. "Man is mortal" would then just mean "all people in general are mortal". You've got that absolutely right. "A man", on the other hand, refers to an adult human being who is a male. A woman would be an adult human being who is a female.

It used to be perfectly normal to use man or men to mean human beings (there was a jokey saying 'Man embraces woman', meaning that the term 'man' was assumed to include women). As a schoolgirl in the 1960s I was corrected (by a female teacher!) for using humans instead of man in a biology essay.

Today, with the rise of ideas about gender equality, it is no longer considered acceptable to use either man or men in this sense.

"Man is mortal" is the best if you do not add anything to it. "Men are mortal" is fine too (but not as good, to my ear).

But, if you add something to it, Man is not as good
. "Men are mortal, gods are not" (ignoring the fact that some gods can be killed)
. "A man is mortal, a god is not"
. "Man is mortal, God is not" would work
. "Man is mortal, {a god is not, gods are not}" is not a parallel construction, so is not as good

You would really only use "The men are mortal" to refer to a specific group of men, but it isn't really necessary with mortal, since all men are. Compare to "The men are warriors"

I appreciate the point Kate raised about the use of men excluding women.