They are indeed synonyms. I would say that there is a slight difference due to the fact that "flourish" and "flower" share the same (French) root. When I see the word "flourish" I tend to think metaphorically in terms of a garden. For that reason the word works very well to describe living things that are doing very well. "Thrive" can also be used for that purpose, but it doesn't have that "flower" feeling. For this reason, while it might sound good to say that a new business is "thriving", it sounds less good (to my ears, at least) to say that a business is flourishing. Businesses just don't seem very flower-like to me, unless of course they are flower businesses. If the local flower store were doing well, I would not hesitate to say that it was flourishing.

Yes, fine.