What’s your favorite kind of weather? The question really is what light is the best for photographing animals? The Chamber of Commerce kind of day where the sun is bright, the sky big blue and the birds are singing is not always the best for photography (unless you are shooting for the CC Bright sunny mid days are often the harshest of light. There will be the highest level of contrast between light and shadow. And, most animals will be deep in the shade.
Contrast that with a spotted clouds. The clouds filter the bright sunlight, softens the contrast and also the shadows. On a partly cloudy day you can see this effect as the clouds move in front of the sun. Wait a second or two for the clouds to move away. See the difference?
Overcast days, cloudy but bright, are ideal for shooting detail and color saturation for everything from feathers to fur. The sunlight is bouncing around in the clouds and is essentially coming from every direction but it is diffused and soft. There will be very little shadow or contrast. My favorite kind of day!
Heavy overcast, on the other hand, is generally very difficult. Light quality is so poor that your ISO may have to be boosted to get the kind of shutter speeds you need.