Are there 'cohomology' functors that respect all Eilenberg-Steenrod axioms except homotopy invariance? What goes wrong in the axiomatic definition of a generalized (co)homology theory if one drops the axiom of homotopy invariance i.e. that homotopic maps should induce the same map in (co)homology?
Or do we have examples? Are there "interesting" or "useful" functors $\mathfrak{h}^{\cdot}:\mathrm{Spaces}\to \mathrm{Ab}$ that respect all Eilenberg-Steenrod axioms except homotopy invariance? Could such an $\mathfrak{h}$ be used to distinguish between two homotopy equivalent non-homeomorphic spaces?
(Take your favourite definition of admissible spaces)
 A: For a manifold $X$, define $H_0(X)$ to be the direct sum of all the tangent spaces to $X$.  This extends in the obvious way to a functor on the category of manifolds and smooth maps.  For a pair $(X,A)$, define $H_0(X,A)=H_0(X)/H_0(A)$.  For $i>0$, set $H_i(X,A)=0$.
This would seem to satisfy all of the Eilenberg Steenrod axioms except homotopy.  
(For cohomology, use cotangent spaces.)
A: Fix an abelian coefficient group $B$.  Given an inclusion of spaces $A \subset X$, you can let $H^n(X,A) = 0$ for $n \neq 0$, and let $H^0(X,A)$ be the set of all (possibly discontinuous) functions from the underlying set $X^\delta$ of $X$ to $B$ which restrict to zero on $A$.  In particular, $H^0(X)$ is the group of all functions $X^\delta \to B$.
(If you like, the map $X \mapsto X^\delta$ is a functor from spaces to spaces which preserves inclusions, "excisive contexts", and takes a point to a (weakly) contractible space, but it does not preserve homotopies.  If you have another such functor, you could compose it with cohomology with coefficients in $B$ and get another example.)
This is a little less silly than it sounds.  The dual homology functor takes $X$ to the set of formal sums $\sum b_x [x]$ of finite sums of elements of $X$ with coefficients in $B$, and similarly for the relative version.  This, as stated, just produces an abelian group.  However, there is a natural topology that can be imposed, and (for CW-complexes) the resulting topological abelian group has homotopy groups naturally isomorphic to the singular homology groups of $X$ with coefficients in $B$.
A: ordinary differential cohomology roughly speaking satisfies the demands of the question
the homotopy axiom fails and is replaced by a variation statement
it is a functor defined for smooth manifolds and smooth maps
the suspension isomorphism is  only true in a weaker version
and there are  [single space] axioms
Simons& Sullivan first issue of topology
finally it is quite geometric and useful in differential geometry & quantum field theory
it extends to  the generalized cohomology context 
essentially by dropping the homotopy axioms
thus it pervades a natural class of examples to the spirit of the question
this question is not obviously a good one, but it is and I salute the questioner.
