Both Panasonic G9 and Sony H300 have 20.0 MP resolution sensors but Panasonic G9's sensor is Four Thirds (17.3 x 13 mm ) and Sony H300's sensor is 1/2.3-inch (6.17 x 4.55 mm ).
Since Panasonic G9'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 Sony H300.
Another difference between these two cameras is that Panasonic G9'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 G9 and H300 sensor size comparison.
As seen above, Panasonic G9 has a 8.0x Larger sensor area than Sony H300. 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.