Nikon 1 V3 has a 18.0MP 1" (13.2 x 8.8 mm ) sized CMOS sensor and features Expeed 4A 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 18MP more than Nikon 1 V3'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.
One other difference between these two cameras that is worth mentioning is that Nikon D810's sensor doesn't have an 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 1 V3 and D810 sensor size comparison.
As seen above, Nikon D810 has a 7.4x Larger sensor area than Nikon 1 V3. Larger sensors give photographer more control on the depth of field and blurry background compared to smaller sensor when shot in 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. Nikon 1 V3 and Nikon D810 sensors have been tested by DxO and the results show that D810 has a better overall score of 97, 45 points higher compared to 1 V3's score of 52.