Like Dan said, society is more higher-level, i.e. at the country, state, or 'people' level. Community is smaller scale and may refer more to a town or neighborhood. People also use community when discussing their church community or a group of people that they are affiliated with due to voluntary involvement in some kind of organization. The society level is not voluntary, generally, whereas a community has more choice as to whether or not they want to join or be a part of it. Society-level is more faceless and you belong moreso by the virtue of your citizenship, birthplace, etc. Community, would more likely be used to describe a group that may actually know eachother or have many common interests.

In ordinary conversation, if I say "our society," I mean the United States. If I say "our community," I mean my town. Each word has more than one meaning, they are all vague, and they overlap.