Sony RX1R has a 24.0MP Full frame (35.8 x 23.9 mm ) sized CMOS sensor . On the other hand, Sony RX1R II has a 42.0MP Full frame (35.9 x 24 mm ) sized BSI-CMOS sensor and features BIONZ X processor.
Sony RX1R II's sensor provides 18MP more than Sony RX1R'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.
Another similarity between these two cameras is that both Sony RX1R II and Sony RX1R sensors lack anti-alias (Low-Pass) filters. 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 RX1R and RX1R II sensor size comparison.
Sony RX1R and Sony RX1R II have almost the same sensor size so none of them has any significant advantage over other in providing control over depth of field when used with 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. Sony RX1R and Sony RX1R II sensors have been tested by DxO and the results show that RX1R II has a better overall score of 97, 6 points higher compared to RX1R's score of 91.