Question: I'm learning Elixir, and the tool 'dialyzer' lets you do static analysis - annotate the function definition with the type specification of the parameters it expects and the output it returns. It's completely optional, but if it were to be used to the full extent possible, how does it match up to good 'ol static typing?

Answer: 
My impression was that dialyzer is not as exact as static typing, meaning that it sometimes doesn't report an error, although it should.

On the plus side, if a dialyzer complains, it's almost always my fault. More often than not, errors are usually due to incorrect typespec.

So, while I don't think dialyzer is as good tool as static typing, it still helps. In particular, I find typespecs very useful, since they can serve as a documentation. Recently I've switched job, and the project I joined is a complex Erlang project. Owing to typespecs it was easy to find my way around the codebase.

So my advice is to use typespecs in larger projects. We write them only for exported (public) functions and records, and it's a big help, without taking up too much time. I usually first make the code work, and when I'm happy with it, add specs, and run dialyzer to verify all is fine.
While static typing takes care of a whole class of bugs, static analysis tools like dialyzer can tell you a lot more about potential pitfalls in your code. Assuming you used specs to their fullest extent, dialyzer would probably be more useful than static typing would be on it's own, at least compared to languages like Go, C#, etc. Something with a much more powerful type system like Haskell still can benefit from static analysis, but less so than a language with a more naive type system like Go. Static analysis is most useful when combined with a static type system though, and since Erlang and Elixir are both dynamic languages, static analysis can only do so much. That said, dialyzer is extremely powerful and useful, and if used consistently, should offer at least the same level of protection, if not more, as the type systems you are probably already familiar with.

I would take a look at the dialyzer docs (<http://www.erlang.org/doc/man/dialyzer.html>), they can tell you a lot more about what you can expect from the tool with respect to Erlang and Elixir. Hopefully that helps!