"They want nobody's sympathy" vs "They don't want anyone's sympathy"

Which is better?

They want nobody's sympathy.

Or

They don't want anyone's sympathy.

I know both are grammatically correct, but I think they should be used for different occasions. One means something slightly different. Am I right?

Also, is there any word like 'no-one'?

The second is preferable. The first is grammatical, but sounds odd (AmE ear).

I cannot think of any occasion where it would be better to say "They want nobody's sympathy," except in some literary or poetic context, for rhythm or rhyme reasons.

But it might be mistaken as "want for" (as in "he wants for nothing", meaning he has everything he could ever want); thus [wanting for] nobody's sympathy could conceivably be misconstrued to mean he had everybody's sympathy.

Or one might also say "They want sympathy from {nobody/no one}." (note: no hyphen in "no one")