They are synonyms. I can't think of any different shades of meaning. ahdictionary.org, a dictionary of U.S. English, shows both spellings, "ambiance" and "ambience." For some reason, it became fashionable in the United States to use the word "ambience" starting, it seems to me, around the 1970s or 1980s. It is a slightly "elevated" word, perhaps because of its French origin. It's a word I'd expect to hear describing an expensive restaurant. McDonalds has an "atmosphere," La Bernardin has an "ambience." Or ambiance.

They basically have the same meaning, but 'atmosphere' is used much more. You might see the word 'ambiance' in promotional literature, or in a tourist brochure, but it is not generally used in everyday language. For example, you would say that there was 'a good atmosphere' at the party you went to yesterday, but you wouldn't say 'ambiance' - It might seem pretentious or simply strange to use this word in a normal conversation.