An interesting comparison. Just one important difference though if I understand correctly: In Gödel's world you will sooner or later ruin your axiomatic system by adding an axiom that is inconsistent with one or several of your earlier axioms. There's no way to know (from within the system) when that happens, except perhaps that it gets a whole lot easier to prove weird stuff. :)
I guess I should have said that you pick as your new axiom a statement that you couldn't prove and also couldn't disprove. Though I'm not sure if even that is the same as logical independence, which is what you really want.
