They have a lot in common syntactically, semantically, and philosophically. They are both languages designed to promate safety and robustness. They're both suitable for writing highly concurrent, massively scale-able applications. The future is lots of cores, and the Rust community is on board with this.

I'd also like to point out that at both of my jobs, we used Erlang for uses cases that are most heavily associated with the language, and for which popular open source alternatives in Elixir/Erlang already exist. We have had better success scaling, operating, and hiring for Rust. We've saved man hours, time, and money.

So I am saying that Rust is a replacement for Erlang. It's not a tool for a different problem.

Early Rust was much closer to Erlang, but it lost more and more of it over time.