At the risk of drifting from the OP's intent, a better comparison would probably be the Fallschirmjager around the time of Crete to the Army's Airborne after North Africa or Sicily. In that regard I'd give it to the US, but not by much.
Plus, you have to stipulate a timeframe for the Waffen SS. Those that formed the core of the SS divisions (which started out as battalions, then Regiments) went through a very rigorous selection process. It has been some years since I read up on it, but the "pipeline" as it were was longer than than the Army Rangers. Does that equal "better?" Not necessarily. The Waffen SS also enjoyed the latest gear the Germans could produce whereas our Rangers were more or less equipped the same as a regular rifle BN...and I think it was the SS who started pushing MG42's down to the squad level but my memory is fuzzy there.
Getting back to timeframes, by '44 the SS was decimated on the Eastern Front. While units were still higly combat effective even at 50% or less strength, the core of the highly selected and trained men were dead whereas our Rangers in 44-45 were really hitting their stride. See also Point Du Hoc and Cabanatuan.
So, if you were to say 1942 I'd give it to the SS, but in 44 the guys with the Sunoco patch are clearly on top. If you paint with a broad brush for the whole of WWII I'd give it to the Rangers by a nose.
But it is all subjective to timeframe, criteria, etc.