Necessity of postulating an empty set

Question

I think the title summarises the question well. In most treatments of ZFC, why is an axiom postulating the existence of an empty set included? Is the Axiom of Infinity ("There exists a set $\mathbb{N}$ such that there exists $\emptyset \in \mathbb{N}$ such that $\forall x, x \notin \emptyset$ and $\forall n, n \in \mathbb{N} \rightarrow n^+ \in \mathbb{N}$") for some reason insufficient?