In sociological terms, the major difference between a community and a society has to do with their respective sizes.
A society is a much larger group than a community. A society can be defined as a group of people that share a defined territory and who have a shared culture. They do not, of course, have to share every single aspect of their culture, but they have to have the same general culture. Therefore, we would typically say that Americans form a society because we occupy the United States and because we generally share an American culture.
By contrast, a community is rather a small entity in sociological terms. A community can only consist of people who typically interact with one another. Two people cannot be part of a community if they do not tend to interact with one another on a relatively intimate basis. Therefore, a community is something whose size is limited by the ability of people to interact with one another on a personal basis.
We can say, then, that a community is a much smaller and more intimate grouping than a society is.