One reason for investigating other consistency proofs (related to that given by Arthur Fischer) is philosophical/foundational. As Arthur Fischer pointed out, we can only prove consistency relative to some other, stronger system. Proving the consistency of $PA$ in $ZFC$ is rather like using a bazooka to open a can of beans; $ZFC$ is much, much stronger than $PA$ and accordingly, you might think, much more likely to be inconsistent. So from a philosophical/foundational perspective, one might want a consistency proof relative to a weaker theory than $ZFC$. Using less to prove consistency ought to improve ones faith that the theory is consistent. In the case of $PA$, Goedel showed that theory equiconsistent with the theory $T$, an extension in finite types of $PRA$ (primitive recursive arithmetic) and thus is in a certain sense finitistically justified.
There is a slightly more technical consideration lurking in the background here. By examining exactly how much one needs to prove a theory consistent, one gets an idea of the precise strength of the theory. Proving $CON(ZFC) \rightarrow CON(PA)$ tells you $ZFC$ is stronger than $PA$, but not by how much. Gentzen's consistency proof for $PA$ using transfinite induction up to $\epsilon_0$ yields an "ordinal analysis" of $PA$ ($\epsilon_0$ being the least ordinal $PA$ cannot prove recursive), and a correspondingly more refined conception of the strength of that theory.