Many people realise that reducing a car's wheel weight leads to a reduction in unsprung weight. Reducing unsprung weight is a good thing, because it makes the suspension more able to react to bumps and changes in the road surface. This means better road holding and grip, and less time in the air when skipping over bumps.
What seems to be less well known is that reducing the weight of the wheels also reduces the rotational inertia of the wheels. In other words, it takes less effort to accelerate and decelerate the wheels. Hence a lighter road wheel takes less effort to accelerate and brake than a heavy one. This means quicker acceleration and better braking.
Another factor is the reduced gyroscopic effect of a lightweight wheel. When a wheel is rotating at speed, it acts just like a gyroscope. This gyroscopic effect of the wheel actually inhibits the steering of the wheel. The heavier it is and the faster it is spinning, the more the wheel resists any steering action. If you want to feel this force for yourself, try holding a spinning bicycle wheel by its axle (hold each side of the axle using both hands) and spin the wheel with your thumbs. Now try to turn the spinning wheel through 90 degrees, so that instead of pointing forwards, it is pointing to the left or right, (by moving your left or right hand forward, and the other one back). You will notice that the wheel itself is trying to prevent the turning action, and the faster you try to turn, the harder the wheel resists. The same is true of car wheels - a lighter wheel is able to turn quicker, with less gyroscopic effect. A light wheel is therefore more responsive to steering input.
One other factor, of why wheel weight is especially important, is that the wheels are, (or should be), at the corners of the car, far out from the centre of the car. If all the car's weight was situated at a point in the centre of the car, the car would have no rotational inertia at all, and it would be able to cope much better with aggressive turn-in and transitions between corners. Lowering the car's wheel weight reduces the weight at the extremities of the car (the outer corners), and so improves turn-in and transition.
Assuming the wheels are strong enough, there is no downside to reducing a car's wheel weight.