There's been a good bit of factory dehorning on the M9A1 frame that Langdon specced for his Elite. The LTT Elite also comes with a Vertec/M9A3 slide; the Elite II came with a heavier Brigadier slide.
Langdon says that the heavier slide isn't necessary or even desirable especially on a gun that could be used for competition; the lighter slide cycles flatter and there's no dip on rechambering like there can be with the heavy Brig slide.
If there were two NIB examples of each, if I were buying it to shoot/use a lot, I'd be getting the Langdon. The Elite II is nice and all, and I'd want it to collect, but I wouldn't run it like I would the Langdon.
FWIW I have a Wilson Brig Tac, and I'm probably going to be moving it and replacing it with an LTT at some point next year..