It's far from the only use case, but my impression from using the Phoenix framework is that anything Node can do Elixir does better (cue
Annie Get Your Gun soundtrack). Elixir is to Node as Ruby on Rails was to PHP in the 'oughts.
It's highly productive, especially at scale, relatively bug-free and easy to test (being a pure functional language with immutable data), designed for (and makes almost trivially easy) extreme concurrency without blocking (even by GC), minimizes the risk of crashes due to errors (YMMV of course), nudges the programmer in the direction of sound and productive architectural decisions, is fast "enough" for most non-number crunching uses (much faster than the 90s dynamic languages, slower than Java/C#/Go/Swift), and, although the ecosystem is still small, it makes up for it with probably the best standard library there is in the form of OTP.
I'd say its problem domain lies in the area of large (large codebase and/or large userbase) realtime apps. Or, more generally, it offers its best productivity relative to the competition for applications where most computations are not particularly complex and/or are repeated, but they may need to be performed at a very high throughput. An excellent server/networking language.