Both features are designed to facilitate the formation of friendships between Diablo 3 players. Clans are limited to 120 members and are invite-only organizations, and you can only be a part of one clan at a time. They're aimed at uniting players who have similar interests, schedules or goals. On the other hand, communities are more casual affiliations; they can be public or private, and membership is "nearly unlimited," according to Blizzard. In addition, you can play in multiple communities at once.
Clans and communities will arrive in Diablo 3 with patch 2.0.1, the update that will be released prior to the March 25 launch of Reaper of Souls. The features are currently available in the version of patch 2.0.1 on the Diablo 3 Public Test Realm and the Reaper of Souls beta.