It seems to me that the most invaluable thing about a static/strongly-typed programming language is that it helps refactoring: if/when you change any API, then the compiler
To start, I'm a native Perl programmer so on the one hand I've never programmed with the net of static types. OTOH I've never programmed with them so I can't speak to their benefits. What I can speak to is what its like to refactor.
I don't find the lack of static types to be a problem wrt refactoring. What I find a problem is the lack of a refactoring browser. Dynamic languages have the problem that you don't really know what the code is really going to do until you actually run it. Perl has this more than most. Perl has the additional problem of having a very complicated, almost unparsable, syntax. Result: no refactoring tools (though they're working very rapidly on that). The end result is I have to refactor by hand. And that is what introduces bugs.
I have tests to catch them... usually. I do find myself often in front of a steaming pile of untested and nigh untestable code with the chicken/egg problem of having to refactor the code in order to test it, but having to test it in order to refactor it. Ick. At this point I have to write some very dumb, high level "does the program output the same thing it did before" sort of tests just to make sure I didn't break something.
Static types, as envisioned in Java or C++ or C#, really only solve a small class of programming problems. They guarantee your interfaces are passed bits of data with the right label. But just because you get a Collection doesn't mean that Collection contains the data you think it does. Because you get an integer doesn't mean you got the right integer. Your method takes a User object, but is that User logged in?
Classic example: public static double sqrt(double a)
is the signature for the Java square root function. Square root doesn't work on negative numbers. Where does it say that in the signature? It doesn't. Even worse, where does it say what that function even does? The signature only says what types it takes and what it returns. It says nothing about what happens in between and that's where the interesting code lives. Some people have tried to capture the full API by using design by contract, which can broadly be described as embedding run-time tests of your function's inputs, outputs and side effects (or lack thereof)... but that's another show.
An API is far more than just function signatures (if it wasn't, you wouldn't need all that descriptive prose in the Javadocs) and refactoring is far more even than just changing the API.
The biggest refactoring advantage a statically typed, statically compiled, non-dynamic language gives you is the ability to write refactoring tools to do quite complex refactorings for you because it knows where all the calls to your methods are. I'm pretty envious of IntelliJ IDEA.