Communities and clans are similar in appearance and function, but have some key differences. Clans are limited to 150 players, and are meant to support gameplay, with announcements for legendary item finds, achievements completed, and easy game joining and party play. Each account can belong to only one Clan at a time.
Communities are a looser affiliation, more of a socializing tool. They offer most of the chat functions of a Clan, but do not support legendary or achievement notifications. Communities allow many more members, thousands vs. 150 for Clans, and a Battle.net account can belong to multiple communities but only one clan.