Rain
The main thing about rain in shogun is that it stops your arquebusiers and musketeers from firing their weapons. This is because the gunpowder gets wet and the fuse can not be lit. Rain also affects the accuracy of your archers - this effect removes almost all the advantages of missile troops.

Hand to hand combat troops are also affected, especially those with heavy armour. Your samurai will become tired much more quickly marching through the rain, and won't be as optimistic about winning the battle.

Your view and that of your enemies can be affected if the rain is heavy, and it is possible to use this cover to try and sneak around for a flank or rear attack.

Snow
Snow also has similar effects on your troops, especially those who are heavily armoured Light and Heavy Wind
Wind affects all the missile troops in the game. Archers are the most affected, even a little wind makes them a lot less accurate. So choosing a windy day when attacking an army with many archers will minimize your casualties. Light wind doesn't cause too much inaccuracy with arquebusiers and musketeers, but you will find them less effective in strong winds.