Sony RX100 III has a 20.0MP 1" (13.2 x 8.8 mm ) sized BSI-CMOS sensor and features Bionz X processor. On the other hand, Nikon P7800 has a 12.0MP 1/1.7" (7.44 x 5.58 mm ) sized BSI-CMOS sensor . Sony RX100 III's sensor provides 8MP more than Nikon P7800'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.
Below you can see the RX100 III and P7800 sensor size comparison.
As seen above, Sony RX100 III has a 2.8x Larger sensor area than Nikon P7800. 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.