I'm not sure if I would be too enthused about real-world-inspired "weather", in the sense that the visual style of this game is so abstract and stylized that 'realistic' rain or snow would just look out of place.
I'd be all for more effects in general, though. It could be anything, as long as it's determined by the music and makes me go "hey, this tune looks cool. I wonder what that other tune will look like".
Currently you (seemingly, anyway) get the same background no matter which tune you pick, just with a set of randomly generated bits and bobs. It would be interesting if - for example - mellow tunes gave calm backgrounds such as distant horisontal shapes and pastel color themes, where fast tunes with a broad base/treble spectrum gave the current more animated background with pulsating abstract shapes and bright neony colors.