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Culture has more historical depth to it (a time factor) and is very, very broad although applying to a particular group. It means the style or styles of that group in virtually every endeavor, especially as these have been handed down over the generations. Includes Art, Teaching, Construction, Schooling, Religion, Cuisine, Dress, Literature, Science, Mores, ....
On the other hand, the word Society has much more to do with governmental structures. Yes, these have some depth in time too, but I think more about laws, norms of behavior, constitutions, solemn compacts, associations, subunits, family, men, women, labor, clergy, clubs, NGOs, city, county, state, commonwealth, common purpose, rules of order, reform objectives -- all things social and having to do with somewhat more abstract concepts than culture does.
But I do admit of some blending as between these two.
Multi-cultural societies exist: USA
Someone can be 'socially uncultured' -- probably describing me at an early age.
Hmmmm. I think this is enough, except to say that a Society can be brand new, as USA once was. But Culture has very deep roots, and is almost never new, unless some tragic event occurs such as in Lord of the Flies, where people have to really start from scratch. Missing a transmission of knowledge and style, a new culture -- IF THEY SURVIVE....