Sony RX10 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, Samsung NX1 has a 28.0MP APS-C (23.5 x 15.7 mm ) sized BSI-CMOS sensor and features DRIMe 5 processor. Samsung NX1's sensor provides 8MP more than Sony RX10 III'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 Samsung NX1'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 RX10 III and NX1 sensor size comparison.
As seen above, Samsung NX1 has a 3.2x Larger sensor area than Sony RX10 III. 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.