All the covers that I think of as “good covers” are the ones that take the original and deconstruct it in some way. A really good cover is one that makes some fans of the original feel that violence has been done. Devo’s take on “Satisfaction” is certainly an example of this; their cover of Nine Inch Nails’ “Head Like A Hole” is another exquisite dance between parody and love of NIN’s carefully-constructed whimper. Siouxsie’s version of “Trust In Me” from Disney’s “Jungle Book” becomes a sexual invitation from Lilith; part of its allure is *knowing* that it comes from a G-rated movie.
If you know the original, it’s always there in the shadows of a cover, and part of the statement the cover is making is in how it positions itself with regards to the original. A bad cover simply replicates, adding little; a good cover takes something and turns it around.
A version of an old standard, on the other hand… a listener might compare it to their favorite take, but it’s just another version, in some way. There may be a Platonic ideal version of SJI encoded somewhere in the universe, but every recording of this unattributed song is just a shadow of it. Some are at obvious angles on flat walls; some are at unrecognizable angles on craggy, complex walls, and by staring at hundreds of them, perhaps we can begin to model the nonexistent object that’s casting them all.
(And then there’s covers that work so damn well that even the original artist says “this isn’t my song any more” – see Dylan and Hendrix with “All Along the Watchtower”, or Trent Reznor and Johnny Cash with “Hurt”. One could say that this is a time where the cover gets closer to the Platonic ideal of the song than the original artist.)
I particularly agree with the first point about covers that deconstruct. I always feel, when I hear such a cover, that the artist has found a completely different song hiding within the song I knew before. An achievement.