Usefulness of Gödel's incompleteness theorem Given a first-order language $L$ and a theory $T$ in that language (a set of formulas of $L$), if $T$ is strong enough to prove arithmetic, then Gödel's second incompleteness theorem tells us that $T$ cannot prove its own consitency, ie $T\not\vdash \text{Cons}(T)$.
Now, imagine for a second that Gödel's second incompleteness theorem is false, and that we do have a formal proof that $T\vdash \text{Cons}(T)$. What confidence does it give us in $T$ ? If $T$ is inconsistent, it proves all formulas and in particular $T\vdash \text{Cons}(T)$. That makes this imaginary proof of $T\vdash \text{Cons}(T)$ fundamentally useless.
So if I rephrase Gödel's theorem tongue-in-cheek, it reads : do not look for a useless proof of consistency, because apart from being useless, it is nonexistent.
Then what makes the incompleteness theorem so famous ? It halted Hilbert's program, but I fail to pinpoint exactly what Hilbert hoped to achieve.
 A: The second incompleteness theorem is useful because it points out an unprovable statement with a more natural meaning than the unprovable statement we get from the first incompleteness theorem.
This can be used as a part of independence proofs: One way to prove that theory $T$ cannot prove some interesting statement $A$ is to show that $T$ proves $A\to\operatorname{Con}(T)$ -- so if $T$ also proved $A$, it would prove its own consistency, which we know it doesn't.
As one example, this is a quick way to know that ZFC (if it is consistent) does not prove that there are inaccessible cardinals. This gives the positive knowledge that if ZFC is consistent, it has a model with no inaccessible cardinal in it, which may be further useful.
If you don't care at all about knowing that this-or-that theory cannot possibly prove $A$, then I suppose this would not count as useful for you. However, if you've banged your head into a wall for long enough trying to prove $A$ -- because that would certainly be useful for your research if only you knew how! -- then you might not dismiss this so quickly.
