Is weather important in D&D?
The answer is a Rule of the Good DM
"Anything is important exactly as much as you want to be important"
I've played and had fun in campaigns without speaking about clouds, rainy or sunny days.
However, the weather can aid you in many features of the game.
Shakespeare's Tempest begins because Prospero creates a storm that causes a shipwreck on his island.
You can use tempests, storms, heavy rains, to move the party in a specific direction or to stop it.
You can use weather to start an adventure: what's behind a big and long unnatural drought? How can heroes stop it?
With weather you can increase variations of your fights.
Heavy rain or fog decrease visibility (penalty to attack rolls or sight check).
Mud causes difficult terrain increasing difficulty of movement or other actions based on dexterity.
Heat causes difficulties to characters donning heavy armor (penalty to attack rolls or movement)
Those are just a few examples, you can find and use many more.
Use weather to have deeper descriptions.
Saying that a lightning crackles in a foggy night while the party is opening Castle Ravenloft's doors, is to give to your players a more vivid description.
Crossing a village (which has just been raided by bandits) under a cold rain makes the place darker.
So, weather adds features in your campaigns and therefore it's useless to determine it day by day
Make a rainy (or foggy, or sunny) day whenever you want, without checking random tables or anything else.