Why does sampling the hemisphere uniformly create such a different picture from sampling points on the light?
Anyone correct me if I'm wrong, but from a past lecture they showed us that if you randomly sample light from any direction you get a noisy result because you will often get 0s from non-light radiance samples. In the right image, they use the probability distribution to only sample directions (or omegas) in the light, so it only integrates over the area of the light. (This uses importance sampling.) Hope that helps!
Adding onto that, for this example specifically, it is also because the light is really small compared to the rest of the hemisphere, so you end up with a lot of pure 0's which result in black dots.