From my own limited experience:

185 tires on 4 inch rims will have slightly better grip, but a slightly 'spongier' breakaway once you exceed their traction limit. 165 tires have lower traction but a slightly 'crisper' breakaway, so are a little more controllable. Very subtle and probably mostly subjective difference vs. a slight but noticable increase in traction.

Of course, the traction and sidewall stiffness if also greatly dependant on the tire itself, the aspect ratio, and the pressure. Lots of variables there.