I *believe* that CGWH spaces were first used in a *systematic* way in the work of Lewis-May-Steinberger on spectra.  It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best reference on CGWH spaces that I'm aware of.  (I haven't looked at the McCord paper Andrey mentions.)

As to why one might prefer to use CGWH spaces, I'm not precisely sure.  But here is one possibility.

A key property of the category of CG spaces is that the product of a quotient map with a space is still a quotient map.  In CGWH spaces, something even nicer is true: any *pullback* of a quotient map (along any map) is still a quotient map.  (I don't know whether this nicer fact fails in CGH, but I suspect it does.)

Another nice fact about CGWH: regular monomorphisms are precisely the closed inclusions, and regular epimorphisms are precisely the quotient maps.  ("Regular monomorphism" means the monomorphism is an equalizer of some pair, and "regular epimorphism" means the epimorphism is a coequalizer of some pair).