It may be easier to think first in the affine case (which in fact is less special than it may first appear, since any closed immersion is an affine morphism).
So suppose that $A \to B$ is a surjective map of Noetherian rings, with kernel $I$, and that $M$ is a finitely generated $A$-module. Then the coherent sheaf associated to $M$ is supported on Spec $B$ if and only if some power of $I$ is contained in the annihilator of $M$.
Now performing $i^{-1}$ in this context (i.e. to a sheaf which is supported on $Y$) does nothing: it is just a "change of perspective functor", which says that a sheaf on $X$ ($=$ Spec $A$ in our situation), which is supported on the closed subset $Y$ ($=$ Spec $B$ in our situation), is the same as a sheaf on $Y$.
On the other hand, performing $i^*$ corresponds, on the level of modules, to passing form $M$ to $B\otimes_A M = M/IM$.
Now there is a natural map $M \to M/IM$, but it will only be an isomorphism if $IM = 0$, i.e. if $M$ is annihilated by $I$.
Thus we have two related, but different, conditions:
The sheaf attached to $M$ is supported on $Y$ if some power of $I$ is contained in $Ann(M)$.
$i^*$ and $i^{-1}$ give the same result when applied to the sheaf $M$ supported on $Y$ if $I \subset Ann(M)$.
This is borne out in Georges example: The ideal $I = T \mathbb C[T]$, and the module $M$ is annihilated by $I^2$, but not by $I$.