They're not equivalent, but they do seem very close together in most contexts when you assume a bivalent (two truth valued) logic.
But they pull apart when it comes to several controversial decisions we have to make in formal semantics and the philosophy of language. Lets consider two prominent examples.
Supervaluationism: This is one solution to the problem of vagueness, which the is problem that many, perhaps even most, predicates don't have determine meanings in terms of their truth. For example, a 7ft man is tall, but precisely where is the line between tallness and non-tallness? Supervaluationism demarcates the LEM from a related rule, the law of bivalence, which says that either P is true or ¬P is true (and thus P is false). On classical logic, this is how the LEM is rendered, because these are the only options possible. But strictly speaking, LEM states that either P is true or is not true. Supervaluationism, and I won't go into detail, says that LEM is true, but that there are "untrue" but not false statements. So, on supervaluationism, the law of bivalence is denied, but excluded middle accepted. If this seems weird, or unwarrented, note I haven't explained the benefits or motivations of supervaluationism at all here. I'm just telling you what it says as an example of where excluded middle pulls apart from how we'd interpret it when learning about propositional logic.
The point of that example is to show that how the Law of Excluded middle manifests in one system is not the full characterisation of it; it's a metalogical property, and has different implications in different systems. So while it could be equivalent to non-contradiction in one system, it might not in another.
Now consider modal conditionals such as "if it were the case that x, then it would be the case that y". Let ">" be the symbol for these statements. Conditional law of excluded middle is then rendered:
"(A > B) V (A > ¬B)".
But consider the following two conditionals (note that Bizet is a french singer and Verdi is an Italian one; i didn't know this when i came across this so was confused:
(BV1) If Bizet and Verdi were compatriots, Verdi would be French.
(BV2) If Bizet and Verdi were compatriots, Bizet would be Italian.
Note that BV2 entails the negation of BV1 and vice versa. It's a very defensible claim that neither one of these are true. And this is the line taken in perhaps the most widely accepted semantics for counterfactuals; Lewis'
Therefore conditional excluded middle fails on Lewis' system. But does the law of non contradiction? No. So they aren't the same thing on some very mainstream systems of conditional logic. For a system that accepts excluded middle for conditionals, see Stalnaker's theory. To learn more about LEwis' system, this is the best guide I've seen: https://math.berkeley.edu/~buehler/Counterfactuals%20Notes.pdf
Warning: this is a technical textbook. But it's rextremely clear.
I hope this clarifies things. The takeaway lesson: don't consider the properties of some logical laws as being identical to their consequences in some system, no matter how ubiquitous that system is. LEM is a metalogical property, and it manifests differently in different logical contexts.