Sony RX100 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, Sony RX1R has a 24.0MP Full frame (35.8 x 23.9 mm ) sized CMOS sensor .
Another difference between these two cameras is that Sony RX1R'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 RX100 III and RX1R sensor size comparison.
As seen above, Sony RX1R has a 7.4x Larger sensor area than Sony RX100 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.