(*) Yeah, I recall rather vividly from last summer that the model, upon which the axiomatic model categories are built, is the following data (in Hovey refered to as the standard model category on Top):
Weak equivalences X -> Y are maps such that the induced maps on the pointed homotopy groups end up being isomorphisms.
Fibrations are Serre fibrations
Cofibrations are what they need to be to fit with these two.

And every homotopy equivalence is a weak equivalence according to this definition. There may, I think, however be more weak equivalences.

***

So, in other words, a natural transformation between two functors really is "just" a homotopy between these two functors, eh?