Nikon D500 has a 21.0MP APS-C (23.5 x 15.7 mm ) sized CMOS sensor and features Expeed 5 processor. On the other hand, Nikon D300S has a 12.0MP APS-C (23.6 x 15.8 mm ) sized CMOS sensor and features Expeed processor.
Nikon D500's sensor provides 9MP more than Nikon D300S'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 Nikon D500'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 D500 and D300S sensor size comparison.
Nikon D500 and Nikon D300S 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
Both Nikon D500 and Nikon D300S sensors have been tested by DxoMark. DxoMark 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. Of the two cameras that we are comparing, D500 has scored 84, 14 points higher than D300S.