Test #3. Same lens, EF 70-200 2.8 II IS, RAW
(1) EOS-M, 125mm, 1/160, f4.5, ISO100, flexizone single, on monopod
(2) EOS-6D, 200mm, 1/1500, f2.8, ISO400, AF, on monopod
I intentionally used a higher ISO on the 6D shot so I could use a faster shutter speed to minimize effects of camera shake on the monopod, so if anything the 6D should have an advantage there.
The white balance was messed up on the shot from the M, but I don't have access to Lightroom or other software to correct it. DPP probably can do this, but I'm not familiar with it.
Oh noes, you are doing it wrong, M should use lens wide open @ f2.8 to match 6D's stopped down DoF @ f4.5.
1/160 vs 1/1500? That's more that 3 stops difference in shutter speed!
Other than that, I think your 6D is messy at pixel level. My previous 1DsII gives pixel-sharp images and looks nothing like the image you posted.
If you are still keen to do comparisons, try those parameters:
M: Av, 100mm, f2.8, ISO 100
6D: Av, 160mm, f4.5, ISO 250
At those settings pointing to the same scene, the shutter speed should be within +/- 0.6ev range.
Remember to turn sharpness settings to middle for both.