0) In Kaplansky's Set Theory and Metric Spaces, he says something like "90% of the time Zorn's Lemma is more natural than transfinite induction. Here is an example of the other 10%." [I don't remember what he proves using transfinite induction -- something about infinite abelian groups, perhaps?]
In my adult mathematical life I have used transfinite induction twice.
I prove that every abelian group is, up to isomorphism, the Picard group of a ring of functions on some elliptic curve over some field (in particular, of some Dedekind domain). The proof uses transfinite induction. Bjorn Poonen explained to me how to remove transfinite induction from the argument. It seems that Bjorn's modified proof is the first proof of Claborn's theorem (every abelian group is, up to isomorphism, the Picard group of some Dedekind domain) that does not use the Axiom of Choice!
2) To show that if $L/K$ is a purely transcendental extension and
$| \ |$ is a non-Archimedean norm on $K$, then it extends to (at least) one non-Archimedean norm
$| \ |$
on $L$. This was an exercise in a course I am teaching this semester:
I wrote it up mostly because I wanted to give a worked example of a proof by transfinite induction. Later, one of the students gave a proof using Zorn's Lemma that I thought was faster and simpler.
Thus in my experience, transfinite induction proofs are few, far between, and can probably be recast in other terms. Nevertheless sometimes transfinite induction is what one thinks to do first, and anyway it's fun to prove something in this way every once in a while.