Although today, an awful lot of folks think they are synonymous, which is where you fall down on your analysis.
If someone doesn't bathe and stinks, and Society ostracises him for it, then that is a societal/cultural action. If the same person becomes a true health hazard to the point where he carries infectious diseases, etc. then gubbermint steps in.
As G.W. once said, Government is organized force. Society is not (or should be not). I oppose the use of organized force to run my affairs in almost all instances. Whereas I might hold my nose and run away from the great unwashed example of this post.
Clear enough? 'Pod