Society is typically a broader term and refers to a large group of people whereas community is more local. Communities are also thought of being closer so that people in them feel more connected to each other. Some ecamples would be
In today's society it is normal to communicate with messenger.
In contrast, In today's community it is normal to... doesn't really make sense.
I feel very connected to my community and care deeply about it. In contrast to I feel very connected to my society... Both sound okay but the first one is much more natural whereas being connnected to one's society is rather abstract.