I think that is a very good example to show one difference.
Here 'association' means people's connection and interaction with one another. Aristotle thinks this kind of interaction cultivates (encourages) the growth of virtue in the citizens.
Organization would not work here. "Political organization" would mean the way in which things are set up, such as how leaders are chosen, who makes certain decisions, and so on. "Political organization" refers to the structure, not the interactions that take place within it. In itself, it would not cultivate virtue.
(Please give us the name of the book. We always ask for the source. )
If organization is not part of the name, we are not likely to refer to it this way. We would say something like, "the organization called Greenpeace," or "Greenpeace, an organization that works to preserve the natural environment." (The organization spells its name as one word.)
Sometimes organization is part of the name, as in The World Organization for Animal Health. In this case, we capitalize the word.