What effects do weather changes have on your character?

Walking around Winterhold, snow and windstorms do decrease the distance in which you can see.

I haven't noticed any heavy impact on move speed, it did feel a bit slower but that might also have been the effect of the current fatigue state and weight of the character and its items.

I haven't seen lower places flooded by rain, but that doesn't impose effects on the character anyway.

None of these effects have an effect on battle, expect for accidentally running into opponents when traveling with a faster horse and not paying attention to your surroundings. Some of the wild monsters I saw around Winterhold were made stronger themselves; so they didn't add weather effects to your character to make it even harder to fight these monsters, I think they only added it to the area to make it more creepy and to denote that you could encounter stronger monsters...

Playing a Khajiit character, I've noticed that I would get a layer of snow on top of my tail when it was snowing. That's the only effect of weather on my character that I've seen so far.