Sony A7 III has a 24.0MP Full frame (35.8 x 23.8 mm ) sized BSI-CMOS sensor and features Bionz X processor. On the other hand, Sony A7R IV has a 61.0MP Full frame (35.8 x 23.8 mm ) sized BSI-CMOS sensor and features Bionz X processor.
Sony A7R IV's sensor provides 37MP more than Sony A7 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.
One other difference between these two cameras that is worth mentioning is that Sony A7R IV's sensor doesn't have an 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 A7 III and A7R IV sensor size comparison.
Sony A7 III and Sony A7R IV have sensor sizes so they will provide same level of control over the depth of field when used with same focal length and aperture. On the other hand, since Sony A7 III has 150% larger pixel area compared to Sony A7R IV, it has potential to collect more light on pixel level hence have less noise in low light / High ISO images.
DxOMark Sensor Scores
DxOMark is a benchmark that scientifically assesses image quality of camera sensors. It 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. Sony A7 III and Sony A7R IV sensors have been tested by DxO and the results show that A7R IV has a better overall score of 99, 3 points higher compared to A7 III's score of 96.