When you look up in the sky you may see the clouds, but when you look at the ground you can see the shadows.
When it rains everything should be grayed a bit and shadows be less sharp. In many videos it seems as if someone opened a firehose in a mid-summer day.
It would suffice to dim the light (much like is done during the night) to a less bright color so that the highlight is much nearer to the shadowed light, if not even the same, removing all shadows.
EDIT: I might also add that when the rain is very strong dark gray haze or fog should be added to simulate raindrops filling up the volume.