Canon 5D MIII has a 22.0MP Full frame (36 x 24 mm ) sized CMOS sensor and features Digic 5+ processor. On the other hand, Nikon D810 has a 36.0MP Full frame (35.9 x 24 mm ) sized CMOS sensor and features EXPEED 4 processor. Nikon D810's sensor provides 14MP more than Canon 5D MIII's sensor, which gives a significant advantage in real life. You can print your images larger or crop more freely.
On the other hand, please keep in mind that Max sensor resolution is not the only determinant of resolving power. Factors such as the optical elements, low pass filter, pixel size and sensor technology also affects the final resolution of the captured image.
Another difference between these two cameras is that Nikon D810's sensor lacks anti-alias (Low-Pass) filter. Removing anti-alias filter increases the sharpness and level of detail but at the same time, it increases the chance of moire occurring in certain scenes.
Below you can see the 5D MIII and D810 sensor size comparison.
Canon 5D MIII and Nikon D810 have almost the same sensor size so none of them has any significant advantage over other in providing control over depth of field when used with same focal length and aperture.