In Elixir, why not use case statements instead of multiple function overloading?

I'm learning Elixir and a little confused as to why we must branch using multiple definitions of the same function, instead of using the case statement. Here is an example from Elixir in Action, first edition page 81, for counting the lines in a file:

defmodule LinesCounter do
  def count(path) do
    File.read(path)
    |> lines_num
  end

  defp lines_num({:ok, contents}) do
    contents
    |> String.split("\n")
    |> length
  end

  defp lines_num({:error, _}), do: "error"
 end 

So we have two defp lines_num instances to handle the cases of :ok and :error. But doesn't the following do the same thing, arguably in a cleaner, and more concise way, and using only one function instead of three?

defmodule LinesCounterCase do
  def count(file) do
    case File.read(file) do
      {:ok, contents} -> contents |> String.split("\n") |> length
      {:error, _} -> "error"
    end
  end
end

Both work identically.

I don't want to learn incorrect idioms as I start out my journey on Elixir so clarifying the downsides of using the case statement in this way, is what I am looking for.