I can't speak for Erlang, but I use Elixir and not primarily for the concurrency model. I just really like the language (and it's functional-but-accessible approach), relative speed (compared to Ruby), fault-tolerance, community.

In particular the fault tolerance and process-based approach is just awesome. I regularly find myself writing code with just the 'happy path' (and only the obvious error handling), which just feels a ton nicer than the more defensive approach I generally take in other languages.

The tooling around the language is also really nice.