Fujifilm X100F has a
24.0MP APS-C (23.6 x 15.6 mm ) sized CMOS X-TRANS III sensor and features X-Processor Pro processor. On the other hand, Sony RX100 III has a
20.0MP 1″ (13.2 x 8.8 mm ) sized BSI-CMOS sensor and features Bionz X processor.
Another difference between these two cameras is that Fujifilm X100F'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 moiré occurring in certain scenes.
Below you can see the X100F and RX100 III sensor size comparison.
As seen above, Fujifilm X100F has a 3.2x Larger sensor area than Sony RX100 III. 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.