You'll notice weight the most on ride quality... Shocks that were designed to handle a stock 15/16 wheel don't do well with heavier wheels. So all motion control is now worse.
The second place you'll notice it is in responsiveness and during transients. Changes of motion and direction, steering feel, etc.
The third place is in acceleration, and the 4th and last is in braking. The effects are most noticeable IN THAT ORDER.
Other factors come into play. The more torque you have, the less it matters on acceleration. The more capable your dampers, springs, and sway bars, the less the effect on ride and handling.
Where the rotating mass is at also matters. If its more towards the center, like lugnuts, it matters less. If its towards the outer, like the tire, it matters more. The tire weighs as much as the wheel. If you stay small, and the tire stays small, the effect is less because only wheel weight has increased. If the wheel weight has increased because it's a bigger wheel, requiring a larger tire, you've doubled the effect or more.
So this is not a black and white question. The worst part generally is what few talk about. It is the "lightness of feet" that is lost. It won't translate to big numbers unless you are on a track, but the car simply feels more sluggish.