There is a notion of Cech homology defined as Jim Conant says, but it is deficient as it does not satisfy many of the properties that one would like with a homology theory. If your spaces are at all general (i.e. more general than polyhedra, simplicial complexes etc) then Cech homology does not have long exact sequences in those situations that one would expect them. It also does not have really nice pairings with Cech cohomology. What goes wrong is that in taking the inverse limit of the homology of nerves of open covers one destroys exactness. (There are well known examples of very simple inverse sequences of exact sequences of abelian groups which when you take their limit end up being clearly not exact. The usual one relates to solenoid spaces.)
If you are just looking at manifolds or similar spaces, then you don't need to be too concerned about these phenomena, BUT if you are working with more general spaces, or with flows, dynamical systems etc. or are working on areas which interrelate with operator algebras, $C^*$-algebras, etc. then there are points to note.
There is a homology theory (Steenrod-Sitnikov homology or Strong Homology) which repairs the deficiencies of the Cech version. The idea can be summed up as saying first take the chains on the nerves of covers then form the homotopy limit of the result, finally take homology, so you replace `$lim H_n$', by $H_n holim$. This has good properties and does relate nicely to the $C^*$-algebra context, e.g. in the well known work of Brown Douglas and Fillmore from the 1970s. (References for Steenrod homology include the lecture notes of Edwards and Hastings, SLN 542, or Sibe Mardesic's book on Strong Homology.)
I'm not an expert, the following is all just guesswork -- I similarly found the original papers unenlightening wrt their motivation.
As you said, the mystery mainly lies in the motivation of the additional step: modding out the functions from $X^{k+1} \to R$ by the subcomplex of functions which disappear on the neighborhood of the diagonal.
First, let's justify looking at neighborhoods of a space. We know from Alexander duality the philosophy of looking at tautness of a subspace $U$ with respect to a space $Y$.
We look at neighborhood $N$ of $U$ in Y (by neighborhood, we mean a subset $N$ of $Y$ that contains $U$ in its interior). The intersection of two neighborhoods of $U$ in $Y$ will be another neighborhood of $U$ in $Y$, so this gives us a system of groups $\{H^q(N)\}$ where $N$ ranges over all neighborhoods of $U$ in $Y$.
For each $N$, this gives us an inclusion $U \in N$, which induces a homomorphism $H^q(N) \to H^q(U)$. The subspace $U$ is said to be "tautly embedded" in $Y$ if this is an isomorphism for all $q$, all $N$, and all coefficient groups. Being taut implies that $U$ is compact and $Y$ is Hausdorff.
This gives us a hint: we are probably modding out by this subcomplex in order to deal with NON compact Hausdorff spaces.
Second, let's justify looking at the diagonal. The diagonal embedding $X \xrightarrow{\Delta} X \times X$, is simply a canonical way to embed a space X into an ambient space endowed with the product topology, $\Delta X := \{(x,x) \in X \times X\}$. It is useful when want to look in the neighborhood of a space $X$ (e.g., at germs of functions on $X$), but $X$ sits in no ambient space. The word, "diagonal embedding," comes from the example of embedding of $R^1 \hookrightarrow R^2$ taking $x \mapsto (x,x)$, that is, taking the line $R^1$ and embedding it into $R^2$ as the line $y=x$.
With this in mind, let's return our gaze to Alexander-Spanier cochains.
Here's my naive guess: modding out functions which disappear on any neighborhood of $X$, $N(X)$, artifically forces $X$ to satisfy the condition that $$H^q(\text{functions which disappear on }N(X)) \simeq H^q(\text{functions which disappear on }X)$$ for all $N$, all $q$, and all coefficient groups. Perhaps modding out by the subcomplex lets us "falsely" satisfy that $X$ is tautly embedded in $X \times X$, so that we may treat $X$ as if it were a compact space.
Below are a few additional comments toward why someone might have thought of modding out by that particular subcomplex.
Establishing notation: $X^{p+1}$ is the (p+1)-fold product of X with itself, that is, for $x_i \in X$, $(x_1, ..., x_{p+1}) \in X^{p+1}$.
$f^p(X) := \{$ functions $X^{p+1} \to \mathbb{Z} \}$, with functional addition as the group operation.
$f^p_0(X) :=$ elements of $f^p(X)$ which are zero in the neighborhood of the diagonal $\Delta X^{p+1}$
If we are examining functions defined pointwise on $X$, it’s natural to look at $X$-embedded in an ambient space, rather than the space $X$ itself. That is, $N(X)$ is the natural home of the jet bundle of $X$.
Functions which disappear on $N(X)$ form a group. If $f$ and $f’$ are both zero on $N(X)$ then $f-f’$ is zero on $N(X)$.
I'm not sure if the following is useful, nor how it fits into the story, but I figured I'd mention it.
The natural home of jet bundles (over a space $X$) is over the diagonal of X. From reading this paper, it seems that Grothendieck brought to the fore the kth neighborhood of the diagonal of a manifold $X$ when he was porting notions of differential geometry into algebraic geometry (this was then ported back into differential geometry by Spencer, Kumpera, and Malgrange). We'll use the standard notation $\Delta X \subseteq X_{(k)} \subseteq X \times X$. The only points of $X_{(k)}$ are the diagonal points $(x, x)$, but, we equip our space $X_{(k)}$ with a structure sheaf of functions, and treat $X_{(k)}$ as if it is made of "k-neighbor points" (x,y) where x and y are the closest points to one another, what Weil called "points proches").
To picture $X_{(1)}$, we might imagine $X$ with an infinitesimal normal bundle, for $X_{(2)}$, an infinitesimal bundle that’s ever so slightly larger of the second derivatives (as we need more local information to take the 2nd derivative), and so on.
If we think of a function $\omega: X_{(k)} \to R$ which vanishes on $X \subseteq X_{(k)}$ as a “differential k-form,” then maybe:
- the functions which vanish to the first order can be thought of as closed forms, $d\omega = 0$,
- the functions which vanish to the second order on the diagonal $X \subseteq X_{(k+1)}$ can be thought of as exact forms for they satisfy $\omega = d\beta$, s.t. $d(\omega) = d(d\beta) = 0$.
Best Answer
An easy example is the torus $S^1 \times S^1$, which has the same homology as but different cohomology ring than the wedge $S^1 \vee S^1 \vee S^2$ (which has no nontrivial cup products).
A more interesting question is whether there are examples which are both closed manifolds. There might be 3-manifold examples but I don't know how to construct them off the top of my head.
For 4-manifolds we can construct examples by finding simply connected closed 4-manifolds with the same middle Betti number $b_2$ but such that the absolute value of the signature is different, which implies that the cohomology rings are not isomorphic. I think we can take $\mathbb{CP}^2 \# \mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$; these both satisfy $b_2 = 2$ but the first one has signature $2$ and the second has signature $0$ (although you don't need to know what signature is to compute that the cohomology rings aren't isomorphic). See this blog post for more background.