Let's look first at the etymology of the two. "Society" comes from O.Fr. societe, from L. societatem (nom. societas), from socius "companion", and its meaning as a "group of people living together in an ordered community" is from 1630s. "Culture", on the other hand, from M.Fr. culture and directly from L. cultura "a cultivating, agriculture," figuratively "care, culture, an honoring," from pp. stem of colere "tend, guard, cultivate, till". It is related to the word "cult", which is derived from the Fr. culte (17c.), from L. cultus "care, labor; cultivation, culture; worship, reverence," originally "tended, cultivated," pp. of colere "to till". As one can see, then, these two words have quite different meanings.
Societies are defined by the patterns of relationships among the people. Cultures are defined by the beliefs, stories, etc. among the people. There can be different cultures and subcultures in a society, but not different societies in a culture. This should give some idea of the relationship between the two. Further, economics is a social science, but it is not a cultural science. Anthropologists study cultures. It is possible to study economics a-culturally (as economists have, mostly, done), to understand it as a social science. But if one is going to understand the details of a particular economy, one has to study the cultural underpinnings of that social organization.
Now, this does not mean that culture is not social. Anthropology is a social science, after all. But culture is but one element of the social. It is an element that gives character to the other elements, that creates the subtle differences in expression among different economies which may otherwise be similar. It is part of society, but it is not identical with society. We can see this in these definitions of culture as
a : the integrated pattern of human knowledge, belief, and behavior that depends upon the capacity for learning and transmitting knowledge to succeeding generations
b : the customary beliefs, social forms, and material traits of a racial, religious, or social group; also : the characteristic features of everyday existence (as diversions or a way of life} shared by people in a place or time
and society as
a : an enduring and cooperating social group whose members have developed organized patterns of relationships through interaction with one another
b : a community, nation, or broad grouping of people having common traditions, institutions, and collective activities and interests
There is certainly overlap in the b definition of society and of culture give above, but the important element is that with culture, it is learned and passed on from generation to generation, which is not necessary for there to be a society. A culture is internalized by the individual members of a society, and that culture informs their actions in many ways. Of course, common culture does help strengthen social bonds, but mere trust is sufficient for that. This goes back to my point that culture in fact underpins society.
So is culture and society two words for the same thing? I think not. I don't doubt that there are many who conflate the two, but these are really two quite different, though related, concepts.