Some people say you should use "succinct" for describing written or spoken words that have already been written or spoken. And that you should use concise more often to describe something in the negative.
"the speech is too long you need to make it more concise"
"wow what a succinct and concise speech you delivered"
Other than this they are pretty much interchangeable, yes you are correct all dictionaries suggest you can swap them.
I think concise is possibly more common and most frequently used. I don't really think it is worth arguing about, people will know what you mean whether you say 'succinct' or 'concise'