If I'm putting on my business hat, I need a reason to use Elixir instead of Rails and that probably has something to do with a better concurrency story, but I'm always curious to hear what other people are getting out of it.
My previous experience with BEAM was using Erlang in a semi-embedded environment and it was fantastic for that. Solid, dependable, deterministic, great response times... all of that. But while we did have a web interface to the system, it wasn't a web site.