I'm not quite certain what Peter May had in mind 40 years ago,
but probably he had in mind the fact that pushouts are a lot better 
behaved in CGWH than in CGH.  Specifically, CGWH is closed 
under pushouts, one leg of which is the inclusion of a closed
subspace.  CGH does not have such nice behavior, and pushouts
like that are used all over The Geometry of Iterated Loop Spaces,
specifically in the construction of a monad from an operad and
in the use of geometric realizations of simplicial spaces.