Independent and undecidable aren't quite the same, even in formal logic. Or rather, sometimes they are but it’s worth being specific.

A proposition P being independent of a theory T means that both (T and P) and (T and not P) are consistent. T has nothing to say about P. This may very well be what Gödel was indicating in his paper.

On the other hand, undecidable has a sharper meaning in computation contexts as well as constructive logics without excluded middle. In these cases we can comprehend the “reachability” of propositions. A proposition is not true or false, but may instead be “constructively true”, “constructively false”, or “undecidable”.

So in a formal logic without excluded middle we have a new, more specific way of discussing undecidability. And this turns out to correspond to the computation idea, too.