There are, as of now, only 2 weather conditions. Rain and sun. You can pick between three routes when starting a journey, the weather type is decided by which route you pick and indicated by a small icon.
When driving through the rain your car will automatically get cleaned slowly. If you brought Uncle Lütfi along, he will repeatedly ask you to turn on your wipers, until you finally do. Turning on the wipers in rain slightly increases visibility. Driving through the rain with standard tyres reduces traction. Alternatively, you can purchase wet wet tyres, made for driving through rain. They will improve your traction. If you drive through sunny weather with wet tyres, they will wear down faster.
When driving in sunny weather, you won't need to worry about traction as much. Your car won't be cleaned as there is no rain, wipers aren't required.