Many uses of Zorn's lemma really only need transfinite induction. For instance, the existence of an algebraic closure of $\mathbb{Q}$ can be done quite simply, without AC.

Many uses of Zorn's lemma really only need transfinite recursion, without any use of AC. Sometimes you don't even need transfinite recursion, but just normal recursion, or even less.

This is especially applicable in specific examples. For instance, you don't need AC to get an algebraic closure of $\mathbb{Q}$.