A perfectly normal space satisfies that all closed sets are $G_\delta$ (countable intersections of open sets). This implies complete normality.
A fully normal space satisfies that every open cover may be refined into an open star refinement. (With Hausdorff, this is equivalent to paracompact.)
All of these properties are implied by metrizability, and imply normality. The pi-Base knows over a dozen fully but not perfectly normal spaces. And it knows four examples that are completely but not fully normal.
What's an example of a perfectly normal space that's not fully normal? Today the pi-Base knows of none.
The question has answers on Math.StackExchange already, but only once you recall that "fully normal" is equivalent to "paracompact" (open covers have locally finite open refinements) assuming Hausdorff.
Then Henno's answer at https://math.stackexchange.com/q/421016 lists a few examples.