Live on is to use someone or something to provide the money or food that you need to live. Live off is to get the money that you need to pay for food, a place in which to stay, clothing, etc. from someone or something.

In the context of giving directions, for example "I live on Main Street," the street they live on is Main Street. "I live off Main Street" means the street they do live on is directly connected to, and it is implied less traveled than, Main Street