Both Canon G1 X III and Sony RX1R have 24.0 MP resolution sensors but Canon G1 X III's sensor is APS-C (22.3 x 14.9 mm ) and Sony RX1R's sensor is Full frame (35.8 x 23.9 mm ).
Since Sony RX1R's has a larger sensor area with the same resolution, this means that it also has a larger pixel area hence better light collecting capacity for a given aperture compared to Canon G1 X III.
One other difference between these two cameras that is worth mentioning is that Sony RX1R'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 G1 X III and RX1R sensor size comparison.
As seen above, Sony RX1R has a 2.6x Larger sensor area than Canon G1 X 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.