Corner turn in is better with the shorter sidewalls. It also understeers a little more with wider rear then fronts (but the throttle still works as a steering aid ). The ride is also stiffer but half of that is certainly the lowering springs. It isn't harsh or uncomfortable, though.
The shorter side wall of the 18 inch tire will cause the car to be a little stiffer. Better put... think of the side wall as a shock. The shorter you make it the stiffer is gets. It will have less give when you hit a bump or pothole and also cause the tire to unload easier when you drive over a bump. A tall side wall will absorb the bump better because it has more room to flex allowing the rim not to move as much. It will help the handling characteristics for the car but on the streets of New York it might hurt it more.