I play hockey, and playing/training is almost exclusively (at any decent level) on artificial pitches. Its very rare that we have issues with the weather, unless its snowing or really cold. The key is how high the pitch is from the water-plane
Some of the older sand-based pitches struggle to drain, and will end up with puddly corners, and puddles in high wear areas, before becoming water-logged. However, the sand content gives them some protection against falling snow.
Water-based drain very well by the very nature of the pitch. However, as said before, if the ground around the pitch is water-logged, then it will struggled to drain. Generally speaking water-based pitches that are heavily flooded, dont stay that way for long once the rain has stopped. Heavy snow is a killer, as it freezes a layer of ice under the snow, the sun cant get to it, and it has to melt from the top down.
Hybrid pitches can be a mixed bag. Some are of very similar type to water-based, and have some form of dressing also.
Generally the majority of sports centres run sand based astro, as thay are cheaper, really hard wearing, and require much less maintenance than any of the other forms.