I see examples of both on the web.
Think You Don’t Need Houseplants? Science Says Different (Forbes)
You say he’s just a friend, but your voice says differently (ScienceDaily)
Life is pain, Highness. Anyone who says differently is selling something. (The Princess Bride (1987))
By the same token, should I say show different or show differently to mean show otherwise?
That research shows different.
That research shows differently.
The first sentence reads like a headline—and newspapers/broadcast media are notorious for abusing grammar to fit things into a headline, to be terse, or to be catchy. If you were writing formally, you may want to say something else—at least use an adverb like “differently”. Also consider that science doesn’t say anything, but scientists speak volumes. Similarly, movie scripts usually depict colloquial speech, often with poor grammar.
Regarding your query about the “… shows different” versus “… shows differently”, consider rephrasing it to “… shows something different” or “… shows a different result”.
Incidentally, Editage.com has a neat article on when to use “show”, “reveal”, or “indicate” in scientific publications.