I think they are synonyms, but come with slightly different connotations. I expect something described as a coat to be relatively heavy and for cold weather, where I expect a jacket to be relatively light and often worn for reasons not connected to how cold it is.
I wrote: "It is quite a new coat" and it was unacceptable. I've read all the comments below plus saw the dropped words for bastante and it says nothing of 'fairly'. I understand the need to think differently for any language but this is not helping one at all.