Panasonic GH2 has a 16.0MP Four Thirds (17.3 x 13 mm ) sized CMOS sensor and features Venus Engine FHD processor. On the other hand, Olympus E-M1 III has a 20.0MP Four Thirds (17.4 x 13 mm ) sized CMOS sensor and features TruePic IX processor.
One other difference between these two cameras that is worth mentioning is that Olympus E-M1 III'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 GH2 and E-M1 III sensor size comparison.
Panasonic GH2 and Olympus E-M1 III 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.