Motivation for "homotopy equivalence" What are some concrete motivations for considering topological spaces up to homotopy equivalence instead of homeomorphism?
I know that most algebraic invariants of spaces (homology, homotopy groups, and so on) depend only on the homotopy type of a space. Are there other situations that suggest that it's better to consider spaces only up to homotopy equivalence?
 A: If you want to know how "homotopy" came up in the first place, you'll have to ask Poincaré, the inventor of the first homotopy group, also known as the fundamental group. The concept of homotopies of paths, i.e. homotopies of continuous functions with domain $[0,1]$, is baked into the very definition of the fundamental group.
Of course you cannot ask Poincaré, so the best we can do is to think about the definition and the theory of the fundamental group, to try to decide for ourselves what role homotopy plays, and why it is a natural/useful/important concept. One reason is quite simple: the fundamental group is useful. It produces a homeomorphism invariant, one which is quite computable, hence quite practical and useful for comparing different topological spaces: non-isomorphism of fundamental groups implies nonhomeomorphism of topological spaces.
If you are not satisfied with that answer, then perhaps my best advice is to learn very, very well all the proofs behind the theory of fundamental groups, and to learn exactly how "homotopy of paths" and homotopies of other kinds of continuous functions are exploited and applied in the development of the theory.
For example, many calculations of the fundamental group are aided by applying the statement that the fundamental group is a homotopy invariant. One of my favorite is that for any connected 2-manifold $M$, if $P \subset M$ is a finite nonempty subset then the group $\pi_1(M-P)$ is a free group. The proof of this theorem is to show that $M-P$ is homotopy equivalent to a graph, which involves constructing various homotopies of maps defined on $M-P$ and on graphs.
For another example, one way to study a topological space $X$ is to study its universal covering space $\widetilde X$ (and the deck action of the fundamental group $\pi_1(X)$ on $\widetilde X$). You can often learn a lot more about $X$ itself by studying not just the homeomorphism invariants of $X$ itself but also homeomorphism invariants of $\widetilde X$. An example of this is that each of the higher homotopy groups $\pi_n(X)$ (for $n \ge 2$) is isomorphic to $\pi_n(\widetilde X)$ (which is proved by constructing lots of homotopies of maps defined on spheres). So if, for example, you discover that the universal covering space $\widetilde X$ is contractible (for example if $\widetilde X$ is homeomoprhic to Euclidean space of some dimension) then you immediately reach a very strong conclusion for $X$ itself: every one of its higher homotopy groups $\pi_n(X)$ is trivial.
What I am trying to get across here is that the "naturality" of the homotopy concept arises either from sheer genius (in the case of Poincaré) or from a deeper understanding of the theory and its applications.
