Ricoh GXR Mount A12 has a
12.0MP APS-C (23.6 x 15.7 mm ) sized CMOS sensor . On the other hand, Ricoh GR III has a
24.0MP APS-C (23.5 x 15.6 mm ) sized CMOS sensor .
Ricoh GR III's sensor provides 12MP more than Ricoh GXR Mount A12'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 Ricoh GR III'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 moiré occurring in certain scenes.
Below you can see the GXR Mount A12 and GR III sensor size comparison.
Ricoh GXR Mount A12 and Ricoh GR III have almost the same sensor size, so neither of them has any significant advantage over the other in terms of providing control over depth of field when used with the same focal length and aperture.