Moss takes its water in across its whole surface, so what is in that water matters more than it would for a rooted plant. The short version: rainwater is kinder than the tap, especially where the tap is hard.
Why rainwater suits it
Rainwater is soft, slightly acidic and free of additives, which is close to what most mosses meet in nature. Hard tap water is alkaline and full of dissolved lime, and over time that lime raises the surface pH and leaves a chalky scale that many mosses dislike, slowly turning a healthy patch tired and grey. Chlorine and chloramine, added to tap water to keep it safe, do moss no favours either.
When the tap is fine
If your water is naturally soft, the tap is perfectly usable. Where it is hard, save rainwater for the moss if you can; a water butt is more than enough for a lawn, a wall or a clutch of terrariums. Letting tap water stand overnight lets some chlorine off but does nothing about hardness.
How to water
Little and often beats an occasional soaking. Misting keeps the surface damp without waterlogging, which is what moss wants; standing wet with no air encourages algae and rot instead. Early morning or evening is best, so it is not drying in the midday sun. For establishing new moss, mist daily for the first few weeks, as in the growing guide; after that, in a shaded spot, the weather does much of the job.