Both Sony A7 III and Sony A7 II have Full frame sized 24.0 MP resolution sensors so sensor size and resolution is not a differentiator between these two cameras.
Below you can see the A7 III and A7 II sensor size comparison.
Sony A7 III and Sony A7 II have almost the same sensor size, so neither of them has any significant advantage over the other in terms of providing control over depth of field when used with the same focal length and aperture.
DxOMark Sensor Scores
Both Sony A7 III and Sony A7 II 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, A7 III has scored 96, 6 points higher than A7 II.
Sony A7 III
Sony A7 II