I'm interested in more than just the use cases (although still curious to hear why someone chose Elixir, whether it be for the language, BEAM, or whatever.) I'm looking for stories of why it was chosen over another language and what the end results were.

Well neither will win any speed races, but if your problem requires soft-realtime functionality, concurrent execution, or incredible fault tolerance you'd do much worse than to choose Elixir/Erlang. And a simpler learning curve than many FP languages (in my experience, at least) sweetens the deal.