Panasonic S1 has a 24.0MP Full frame (35.6 x 23.8 mm ) sized CMOS sensor and features Venus Engine 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 12MP more than Panasonic S1'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 similarity between these two cameras is that both Nikon D810 and Panasonic S1 sensors lack anti-alias (Low-Pass) filters. 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 S1 and D810 sensor size comparison.
Panasonic S1 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.
DxOMark Sensor Scores
DxOMark is a benchmark that scientifically assesses image quality of camera sensors. It scores camera sensors for color depth (DXO Portrait), dynamic range (DXO Landscape) and low-light sensitivity (DXO Sports), and also gives them an overall score. Panasonic S1 and Nikon D810 sensors have been tested by DxO and the results show that D810 has a better overall score of 97, 2 points higher compared to S1's score of 95.