Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic. [...] The original (second-order) Peano axioms... have ONLY ONE MODEL, up to isomorphism." (my emphasis)
Would this be an example of the incompleteness of the first-order Peano Axioms?
Update 5 months later
Though I don't fully understand Noah's thoughtful answer and am at a loss to formulate any more follow-up question(s), as helpfully suggested, I did find his comment that Gödel's Incompleteness Theorem has "nothing whatsoever" to do with 2nd order Peano arithmetic to be very interesting. On this basis, I have accepted his answer.
No. Compactness merely guarantees that $\mathsf{PA}$ (as is now standard I'll write simply "$\mathsf{PA}$" for first-order Peano arithmetic) has multiple models up to isomorphism. It does not prevent such models from all looking the same from the perspective of their internal first-order theories - that is, from being elementarily equivalent. Indeed, compactness applies to all first-order theories, including the complete ones. True arithmetic $\mathsf{TA}$ for example has $2^{\aleph_0}$-many non-isomorphic countable models, but is by definition complete.
The incompleteness of $\mathsf{PA}$ is a much stronger result, and does not follow from coarse considerations alone. Compactness shows that $\mathsf{PA}$ is not categorical (and indeed no first-order theory with infinite models is categorical). But even that is limited: it takes more than mere compactness to even show that $\mathsf{PA}$ isn't $\aleph_0$-categorical (we have to bring in Lowenheim-Skolem, or use Godel).