Are there things Elixir can do that Erlang cannot, or vice versa?

You shouldn't stumble on anything you can do in one that you can't in the other, since you can freely call Elixir code from Erlang and vice-versa. You can even easily mix Erlang and Elixir files in one project.

In Elixir:

:erlang_module.erlang_function(args)

in Erlang:

'Elixir.ElixirModule':elixir_function(args)

TL;DR - Start with Elixir

Erlang has a steeper learning curve compared to Elixir. Once you start learning Elixir, you will automagically start learning Erlang. Hence, start out with Elixir. Elixir is written in Erlang and Elixir. See distribution on Github (since Elixir is full of macros aka meta-programming).

You can use Elixir with Erlang and vice-versa, hence the full Erlang eco-system of 20+ years of libraries.

More details from Erlang Solutions

Elixir’s ‘out of the box’ productivity is accomplished by a strong focus on tooling and convenience in expression of data manipulation. System design is the same in Elixir and Erlang, but Elixir removes a lot of boilerplate code and is easier to extend. The removal of boilerplate raises productivity and allows programmers to get feedback faster – crucial when you want to launch your product to market as fast as possible. Less boilerplate also makes for happy developers, and happy developers are unsuprisingly productive developers.

Joe Armstrong's (Erlang inventor's) blog post about Elixir

Start here to learn about Elixir - Getting Started

Once you feel its going well, work your way towards practicing on Exercism and other resources.