When it begins to rain, flying bugs should either disappear or become grounded. This could also force bees back into their nest, disperse midge swarms, etc.
Rain is great, But i would like some differentiation from clear skies aside from worms, so making flying bugs disperse, find shelter, etc would create that feeling of the world reacting to the rain.
Aside from bug changes (I focused on bugs because I live next to a swamp, too many midges), perhaps it could affect fishing chances as well. If it starts raining, it could incentivize people to go out and fish, or the opposite if it makes fishing harder.
Also, just for fun, make very deep ocean damage increase during rain/storms substantially. did you cross the sea before a rainstorm, and now are stuck on the other side? sucks, wait for it to clear up.