Panasonic LX100 has a 13.0MP Four Thirds (17.3 x 13 mm ) sized CMOS sensor and features Venus Engine processor. On the other hand, Sony RX1R has a 24.0MP Full frame (35.8 x 23.9 mm ) sized CMOS sensor .
Sony RX1R's sensor provides 11MP more than Panasonic LX100'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 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 LX100 and RX1R sensor size comparison.
As seen above, Sony RX1R has a 3.8x Larger sensor area than Panasonic LX100. 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.
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. Panasonic LX100 and Sony RX1R sensors have been tested by DxO and the results show that RX1R has a better overall score of 91, 24 points higher compared to LX100's score of 67.