Sony A7R III has a 42.0MP Full frame (35.9 x 24 mm ) sized BSI-CMOS sensor and features Bionz X processor. On the other hand, Sony HX350 has a 20.0MP 1/2.3 (6.17 x 4.55 mm ) sized BSI-CMOS sensor and features BIONZ X processor.
Sony A7R III's sensor provides 22MP more than Sony HX350'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 Sony A7R III'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 A7R III and HX350 sensor size comparison.
As seen above, Sony A7R III has a 30.7x Larger sensor area than Sony HX350. 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.