5Dmkiii screen vs 6D screen

I use the 5Dmkiii and the 6D side by side pretty often. I've noticed that with the same settings the 6D will often portray a different looking image on the back of the camera with respect to white balance. What gives?

I often find myself wondering if I should make a white balance shift (on the 6D)on account of how the image looks on the back of the camera.

How do they look after you download them to a computer?