Communities and networks (of people) are closely related concepts in that they describe how you connect to other people. Both are important. The differences between community and network are defined by the ways that you connect to others.
What does it mean that the average number of “friends” for someone on Facebook is 338 or that an average LinkedIn user has close to 1,000 connections? While it might sound impressive and even provide a sense of sociability, it is important to realize that these reflect robust networks and, most likely, not communities.