Unlike Mute, Do Not Disturb turns of both sounds and taps on your Apple Watch.
The idea of Do Not Disturb, as the name suggests, is to keep your Apple Watch collecting notifications without actually notifying you. That might sound strange, but if you're asleep, in a meeting, at the movies, or otherwise at a time and place where you don't want noise or haptics to bother you, but you don't want a list of everything you might have missed in the meantime, Do Not Disturb will do exactly that.
Full story from the iMore Blog...