The majority of my experiences are with my 1Ds MkIII's and customer (I print for other photographers) 5D MkII files, and I feel that this is an equally valid, if not more so (and the OP agreed with me), comparison due to the aforementioned same sensor but different output.
I know from hands on experience that 5D MkII RAW files are not as malleable as 1Ds MkIII RAW files.
This is not a question of DR, noise, banding, FPN or any one "issue" it is a comment on the actual finished output of the image file, the 1 series cameras just have more flexibility in them when worked by the same person.
Here is an example of a torture test where a 5D MkII would get very different results, unfortunately I don't have direct comparison shots but try this with a 5D MkII.
Image is a 1Ds MkIII shot at 1600iso and underexposed by two stops. This was then lifted in post by two stops to give an effective 6400iso at an EV of 10ish, or just past sunset light. I did noise reduction to a level many would feel excessive but I wanted to see what detail was left, as you can see from the 100% crop the individual hairs are still rendered well. Note there is zero banding, no noise and no FPN. Detail and DR have been badly impacted but the image is 100% usable at a variety of sizes.