Elixir/Erlang are one among best functional programming languages for web today. Phoenix is a very performant and productive web framework. You can build a typical web app really fast using Phoenix and Elixir.
Well, the erlang vm is what runs Elixir apps. Elixir runs on the erlang VM (called BEAM) similar to how scala/clojure run on the JVM. However, while writing a lot of elixir code you end up using the erlang standard library for things which are not available in Elixir, :crypto is a module which comes to mind. Elixir apps at the end of the day are erlang apps.