I think this is because the density of water is much higher than the density of vapour at the same temperature. Therefore, when water condenses in the atmosphere, there is only enough water available in the immediate neighbourhood to form one little drop.
Clouds are made of little drops of liquid water, too. I think what must happen is that there is a maximum drop size (which must depend on the exact atmospheric conditions at the time, but for a given set of conditions there is always a maximum size) which can stay suspended in a cloud. I’m sure it’s more complicated, but I think of it like this: as the droplet falls it evaporates – if it is small enough then it will evaporate completely before it falls out of the cloud (i.e. not rain).
So, once the droplets in the cloud exceed whatever the minimum size is then they fall out of the cloud as rain. To get a continuous stream you would need a really big drop to form, but there is no way that could stay suspended in the cloud because it is too heavy.