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 A9 has a 24.0MP Full frame (35.6 x 23.8 mm ) sized BSI-CMOS sensor and features BIONZ X processor.
Sony A7R III's sensor provides 18MP more than Sony A9'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 similarity between these two cameras is that both Sony A9 and Sony A7R III sensors lack anti-alias (Low-Pass) filters. 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 A9 sensor size comparison.
Sony A7R III and Sony A9 have almost the same sensor size so none of them has any significant advantage over other in providing control over depth of field when used with same focal length and aperture.
DxOMark Sensor Scores
Both Sony A7R III and Sony A9 sensors have been tested by DxoMark. DxoMark 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. Of the two cameras that we are comparing, A7R III has scored 100, 8 points higher than A9.