Distinguishing factors — In general, communities are fluid groups (large and small) that form around shared values, with blurry lines around who belongs. On the other hand, organizations are structured groups that form around a common goal, with clear lines around who belongs.
Interplay between the two — Communities exist within organizations. Organizations arise within communities. It’s helpful to understand this interplay to develop an effective communication strategy with (or within) any group of people.
Members & leaders — While organization’s tend to rigidly define titles and roles, communities often have fluid dynamics between “members” and “leaders”. Anyone might lead on one issue and follow on another. This dynamic flow of voices helps form the fabric of a community. If a “leader” attempts to control others, they are, almost by definition, no longer a community member — though in the sense of an organization, they may retain their position.