Sony RX1 has a 24.0MP Full frame (35.8 x 23.8 mm ) sized CMOS sensor . On the other hand, Panasonic S1R has a 47.0MP Full frame (36 x 24 mm ) sized CMOS sensor and features Venus Engine processor. Panasonic S1R's sensor provides 23MP more than Sony RX1'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 difference between these two cameras is that Panasonic S1R'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 RX1 and S1R sensor size comparison.
Sony RX1 and Panasonic S1R 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.