Let $S$ be a non-empty partially ordered set with respect to a relation $\leq$. Then:
Zorn's Lemma: If $S$ has the property that any totally ordered subset $U\subset S$ has an upper bound, then $S$ has a maximal element
Ascending Chain Condition: Every strictly increasing chain eventually stabilises.
The a.c.c implies that every subset of $S$ has a maximal element and we can also deduce the same from Zorn's Lemma, so what, if any, is the difference between the two? I feel like they are both saying the same thing