Nikon P100 has a 10.0MP 1/2.3-inch (6.17 x 4.55 mm ) sized BSI-CMOS sensor and features Expeed C2 processor. On the other hand, Sony RX10 III has a 20.0MP 1″ (13.2 x 8.8 mm ) sized BSI-CMOS sensor and features Bionz X processor.
Sony RX10 III's sensor provides 10MP more than Nikon P100'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 P100 and RX10 III sensor size comparison.
As seen above, Sony RX10 III has a 4.1x Larger sensor area than Nikon P100. 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.