The magic is in the supervisor pattern, explained here for erlang: http://erlang.org/documentation/doc-4.9.1/doc/design_princip...
It is hard to describe why this "feels different" in Elixir than it does in Express.js or a Tomcat running a Java application. It's all experiential for me, but maybe I can put the sentiment in words: I always KNOW that whatever part of my application may break, however much and for whatever duration, the scheduler and the supervisors will make sure that the rest of the system runs exactly as intended, and the broken part of the system will be back up eventually. I did not have this feeling (as strongly) prior to working with Elixir.
But I will admit this is a very subjective position. And I am not sure you'd experience it the same way were you in a similar situation.