Why the "W" in CGWH (compactly generated weakly Hausdorff spaces)? In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the category Top topological spaces, in order to do homotopy theory.
The most important difference between CGH and Top is that in CGH there is a functorial homeomorphism $$\mathrm{map}(X,\mathrm{map}(Y,Z))\cong \mathrm{map}(X\times Y,Z),$$
a fact that is only true in Top under the extra assumption that $Y$ is locally compact.

But in more recent papers, I see that people use CGWH spaces instead of CGH spaces...
Why?
Could someone explain to me what goes wrong in CGH spaces
(please illustrate with an example), and explain how the "w" fixes everything?
Also (following Jeff's comment), to whom should the "w" be attributed?
One more wish: can someone give me an example of a CGWH space that isn't CGH?
 A: Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.
Start with a countable metric space X so that with one exception x, each point is open, but so that at the exceptional point, X is not locally compact at x. 
It is easy to find such a subspace of the real line. 
( start with 0 and (1/n)+(1/(m+n))
Now delete each 1/n).
Let Y be the one point compactification, adding to X  a new point y, whose neighborhood complements are compact in X. In the new space Y, compact subsets are closed (and in particular Y is WH), but x and y are inseparable.
See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.
A: To flesh out my comment above: in the Errata to Geometry of Iterated Loop Spaces (p. 485 here: http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf) May states that he should have used weak Hausdorff spaces "in order to validate some limit arguments."  I'm not sure exactly what he means; in particular I would think he really means colimit arguments.
A: In the [following answer]
to a closely related question, Peter May explains a crucial feature of CGWH spaces not shared by CGH spaces.
A: 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  It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best reference on CGWH spaces that I'm aware of.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.  Update: Having looked at McCord's paper, it does indeed seem to be the one to introduce CGWH (the idea of which he attributes to J.C. Moore.))
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.("Regular monomorphism" means the monomorphism is an equalizer of some pair.)  (I originally said here that regular epis in CGWH are precisely quotient maps, but on reflection I'm not sure this is true.) 
A: A web search suggests that the category of CGWH spaces was introduced in the paper "Classifying Spaces and Infinite Symmetric Products" by M. C. McCord (Transactions of the American Mathematical Society Vol. 146, (1969), pp. 273-298). 
McCord motivated introduction of his "weak Hausdorff" separation axiom by noting that

"the requirement of the Hausdorff condition can be a problem
  because certain standard operations on spaces can lead outside the category", 
  in particular quotient spaces in algebraic topology and topological algebra.

