I am working with the following definitions:
A topological space is normal if and only if every pair of disjoint, nonempty closed sets can be separated by a continuous function.
A topological space is called perfectly normal if whenever C and D are disjoint, nonempty, closed subsets of X, there exists a continuous function $f:X \rightarrow [0,1]$ s.t $C = f^{-1}(0)$ and $D = f^{-1}(1)$
I can't see why perfectly normal is stronger than normality, from these definitions they seem equivalent to me.
I lied when I said the two definitions were equivalent; I was overlooking something.
Say $A$ and $B$ can be "separated by a continuous function" if there exists $f$ so that $f=0$ on $A$ and $f=1$ on $B$. Then it's true that a space is normal if and only if any two disjoint closed sets can be separated by a continuous function.
Saying $A$ and $B$ are separated by $f$ says that $A\subset f^{-1}(0)$ and $B\subset f^{-1}(1)$. The definition of "perfectly normal" says $A= f^{-1}(0)$ and $B= f^{-1}(1)$, a stronger condition (people sometimes say that $A$ and $B$ are "precisely separated" by $f$).
It's not hard to show from Urysohn's Lemma that a space is perfectly normal if and only if it is normal and every closed set is a $G_\delta$.