A remote function is a function in a module that isn’t this one.
A local function is one defined in the current module.

There is a slight cost in the BEAM to execute a remote function over a local one, but not material to a design decision.

A remote call is every function call which is qualified by its module.

defmodule M do
  def f(0), do: 0
  def f(n) when n > 0, do: M.f(n-1) + 2
end

The recursive call here is a remote call, with all of its costs.

That is not fully true as __MODULE__.foo() is also remote call. In short the difference between remote and local call is that you put the module name before remote call.

Any remote call need to point to public function.

Though here we have to remember, that for &__MODULE__.foo/0 to work, the function needs to be exposed/public, by using def, while when doing &foo/0 the function is allowed to be private (defp).