ok, Let's look at the statemt that says
For all but finitely many $w \in C$, $f(z)-w$ has $n$ distinct roots in $\mathbb C$.
As written, that statement doesn't say there are any roots at all when $w$ is one of those finitely many exceptions. Thus it falls short of entailing the fundamental theorem, which says there is always at least one root.
You wrote "including multiplicities, $n$ dinstinct roots". Maybe you're missing the meaning of the word "distinct". If any of the roots has multiplicity more than $1$, and the sum of the multiplicities is $n$, then the $n$ roots that you have (including multiplicities) are not distinct.
Regardless of the value of $w$, there will be $n$ roots if you count them by multiplicities, i.e. the sum of the multiplicities will always be $n$. But for finitely many values of $w$, the roots will not be distinct.
You ask whether the statement is "equivalent" to the "fundamental theorem of algebra". If "equivalent" means they are either both true or both false, then they are. If equivalence is to be defined relative to a specified axiom system in which the axioms are not enough to decide whether either statement is true or false, then it can make sense to say what "equivalent" means.
But if "equivalent" means the deduction of either as a corollary of the other is quick and simple, then "equivalent" is not all that precisely defined. Sometimes it means somebody has proved that either both are true or both are false, without saying which.
Obviously if there are $n$ roots, counted by multiplicities, then there is at least one root, and that's what the fundamental theorem says.
On the other hand, if the fundamental theorem is true (and it's not really a theorem of algebra as we usually understand that term today), then a simple theorem in algebra says we can write
f(z) = (z-z_1) g(z)
where $z_1$ is a zero of $f(z)$, and then applying the fundamental theorem and that simple theorem of algebra again, we get
f(z) = (z-z_1)(z-z_2)h(z)
and so on. So you get $n$ roots, counted by multiplicities. You can make this into a precisely stated proof by mathematical induction.