I shudder to call this a cover. Even though Julee Cruise provided the original vocal to this song, it was Angelo Badalamenti's haunting musical arrangement that makes the song memorable. And yes, he and his bands are accompanying here.
But it is a cover, at least of the vocal version. Sky Ferreira is much more of a belter than Julee Cruise, with a higher voice, bringing a slightly different feel to the vocals.
I'm down to the third paragraph and I haven't even mentioned the obvious - this is the theme from Twin Peaks, which is currently experiencing a revival on Showtime, and premiered last night. And yes, both Sky Ferreira (in a non-musical role) and Julee Cruise (as the roadhouse singer) are scheduled to appear. As a huge Twin Peaks fan, I am quite excited.
And when I found this cover, I was excited that everything had come full circle. So enjoy.