This question is in the context of the Beam VM and the capabilities that it provides, not in the general context of what a Turing complete language can do. I want to invest
Just to preface - I have only used Elixir in production and not Erlang.
I would honestly recommend Elixir. This is my opinion and not necessarily the right one for you, but below I will list my reasons why.
Really, if you have a mess around with the two you will probably come to the same decision that Elixir is a much nicer language with all the features of Erlang + More. It's on the rise as well, I can't remember the exact numbers but I remember reading something from the Hex website (package manager) about a considerable increase in package downloads.
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)
There are a few things. I think you can't make a recursive anon function in Elixir. Now to be fair this is something that in 8+ years of Erlang I don't think I have ever needed to do, and if I did it could be done easily in some other way. There are probably a few other things like that that quite honestly you can ignore.
In general, for things that most people care about anything you can Do in Erlang, you can do in Elixir and vice versa. The community on the Elixir side seems to be more active so I would suggest starting there. I have recently moved from Erlang to Elixir and with the exception of a few fancy things in Proper I can say that it is a pretty easy transition
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.