I discovered from someone at the BBC that the only reason they do the Tuesday live edition is a new rule at the BBC stating that they can only show repeats after 11pm (or maybe it's 10.30?), Later's traditional Friday night slot.
So they created the Tuesday live version so that Friday's was then a repeat.
Except that apparently they first film Friday's in full, do any pickups they need to cover mistakes and then do Tuesday's live show after that. So Friday's show isn't really a repeat at all, it's just the same people doing the same bits and playing some of the same songs. Even the interviews are different.
Anyway, that means if a band's playing the show that you like, you're technically going to see two different performances of the same song over the course of each show.