Is it dark enough to spawn monsters, or bright enough to make them despawn?
No. Rainy weather reduces day-time light-levels to 12 at mid-day, which is not low enough to spawn monsters. This 3-point decrease does mean that, as twilight falls, it will be darker sooner and monsters will spawn just that bit earlier—but of course, it's not "day" anymore at that point.
However! Thunderstorms have a light level of 7 at mid-day, meaning that monsters will spawn then. That's just one more reason to not go walking around when lightning is crashing down.
Notch said on twitter:
MUOAHAHA! Thunder Storms lower the brightness. That means daytime monsters during storms. This is freakily atmospheric.
So that means monsters will spawn during a storm (thunder and lightning) in the day, but not when it is only raining.