You would expect the 5DIII to perform better than the 6D being that the latter is what I consider the Rebel of full frame
"Better" is relative here - the 6d gains a little higher iso capability (maybe about 1/3 stop) but produces less sharp images than the 5d3, so it's about what people want from a ff sensor. To me, the 6d appears like a downgraded 5d3 sensor with some readout adjustments and forced chroma noise reduction even in lower iso modes (in higher iso modes, the 5d3 also has nr in raw).
Can you show me where the mk ii beats the mk iii?
The 5d2 has a higher dynamic range (highlight recovery) and spatial resolution (i.e. sharpness) in low iso modes, even with the little less mp count than the 5d3. This fact has been known for long, Canon changed the iso/mp/dr tradeoff to gain better high iso performance and probably less banding on the 5d3 - all these ff models basically all use the same sensor tech. Since ff is about higher iso shooting it's a smart move, but I recon most 5d3/6d owners don't really want to know anyway :-p
The best proof of this I know is a recent German 6d review one on traumflieger.de - but the differences between the 3 ff bodies are not large enough to make a decisive difference for potential buyers, but it proves the point.