For me, weather adds variety to the game, not just in gameplay but in my visual experience. Changing seasons adds a stronger sense of time passage, especially when I'm playing with aging set at my choice not the game's. I'm dearly hoping for world differentiation in weather too, to add more richness to the world differentiation. And changing weather makes one same-old Sims day different from another same-old Sims day when I'm playing at a fairly mundane level, even if it's just doing that in the background. That affects my experience too, and makes it richer.
