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: 
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!
For one thing static typing is built into the compilation phase--sort of impossible to miss. Static analysis on the other hand is something that a developer has to run voluntarily.

Likewise, one doesn't have to annotate anything in Elixir; it's entirely down to programmer discretion. In statically typed languages it's impossible to avoid.

I would say your question is sort of broad and therefore hard to answer with any rigor. You might want to put this over on Programmers.Stackechange.