I just realized I have always taken the homotopy category of spaces to be
CGWH spaces inverted wrt weak homtopy equivalences. i.e. I am considering the homotopical category $(CGWH, whe)$, and $CGWH[whe^{-1}]$
So how is this category compared to
the CGWH spaces inverted wrt homotopy equivalnces? i.e. $(CGWH,he)$ and $CGWH[he^{-1}]$?
I know if we replace CGWH spaces by CW spaces, then the latter two are equivalent.
The best description is simply that it is the category of spaces where we quotient the morphisms by homotopy. Strom has a paper "The Homotopy Category is a Homotopy Category" where he shows that ordinary fibrations and cofibrations give a model structure. In particular, every object is fibrant and cofibrant which essentially tells you this is the best possible description, as opposed to spaces with weak equivalences inverted which often is better described as CW complexes where we quotient by homotopy.
In particular, there is a functor from spaces with homotopy equivalences inverted to spaces with weak equivalences inverted that respects the respective equivalences. This does not induce an equivalence of homotopy categories because there are spaces (like the double comb space) which have trivial homotopy groups, but are not homotopy equivalent to a point.