In mountain locations, rain tends to run off the ground and into rivers as it rains. This depends on the region, but happens often in Spring/Fall.
Snowmelt, when the weather warms gradually, will slowly melt. It will seep into cracks and replenish underground water sources. It will also slowly replenish rivers often into (and sometimes throughout) the Summer.
So, a gradual snowmelt helps with the water cycle.[1] If warm weather comes fast, then the benefits over rain are diminished. Then less desirable effects like flooding can happen.
Milder winters mean huge swathes of the world where certain disease-ridden and economically disastrous pests like ticks and termites are currently walled out from due to the extreme cold in the winter are suddenly going to be much worse places to live.