To some extent it doesn't really matter, as most people only know the most popular or recent version, but if you give out the original artist info when you announce the song, you may get some people interested in going back to listen to the original version and expanding their horizons. I knew Shaky's Green Door wasn't an original, but I didn't know who originally did it or what it sounded like until I looked it up just now.

I've just looked up another one The Horses - Daryl Braithwaite, and that was originally a 1989 song by Rickie Lee Jones and written by her and Walter Becker (of Steely Dan fame). But it was obviously the Daryl Braithwaite version that most Australians would know being a no.1 there in 1990.