Homotopy groups other than $\pi_1$ : what are they good for? Homotopy groups $\pi_k$ were introduced before the homology groups $H_k$ in the 1st topology book I read and the 1st topology course I took. Later on $\pi_1$, $H_k$, and $H^k$ appeared in numerous contexts in a variety of subjects. 
On the other hand, I never encountered $\pi_k$, $k\ge 2$ beyond the topology textbooks, with the exception of the $\pi_3(S^2)$ curiosity, which IMHO is just an algebraic sideshow for Hopf fibration.
Therefore the question: are homotopy groups $\pi_k$, $k\ge 2$ useful for something that cannot done with other invariants such as $H_k$?
 A: I think, they have an intrinsic appeal: If one cares about manifolds, one should care about maps between manifolds. As maps between spaces appear so often in topology, of course also homotopy groups do. But some concrete examples:
1) Thom related the study of bordism classes of manifolds to the (higher) homotopy groups of Thom spaces. [Later this was rephrased into homotopy groups of Thom spectra.] This allowed him to use methods from algebraic topology to completely classify manifolds up to bordism.
1b) In particular, framed bordism classes are in relation to (stable) homotopy groups of spheres. These in turn are, by Kervaire-Milnor, in close relation with exotic differentiable structures on spheres. 
2) The elements of the homotopy groups of $SO(n)$ are in one-to-one correspondence with isomorphism classes of vector bundles on spheres. 
3) An example for the answers of John and Ryan: The Poincare conjecture tells us that every closed manifold homotopy equivalent to the sphere is actually homeomorphic to the sphere. How to test whether a manifold $M$ is homotopy equivalent to the $n$-sphere? It is enough to show that $\pi_1M = 0$ and $M$ has the homology of the $n$-sphere. Indeed, by the Hurewicz theorem it follows that $\pi_nM = \mathbb{Z}$. Thus, we can find a map $S^n\to M$ inducing an isomorphism on homology. Thus, it is a homotopy equivalence (by a version of the Whitehead theorem).
4) It is very useful to identify a space as an Eilenberg-MacLane space. An Eilenberg-MacLane space of type $K(G, n)$ is per definition a space $X$ (homotopy equivalent to a CW-comples) such that $\pi_nX \cong G$ and all other homotopy groups are zero [thus, its very definition uses homotopy groups]. The main use is the following: Homotopy classes $[Y,X]$ for $X$ a $K(G,n)$ are in one-to-one correspondence with elements in the cohomology $H^n(Y;G)$. 
For example, $\mathbb{CP}^\infty = BU(1)$ is a $K(\mathbb{Z},2)$. Thus, we can immediately conclude that homotopy classes $[Y,BU(1)]$, i.e. isomorphism classes of line bundles on $Y$ (if $Y$ is paracompact), are in one-to-one correspondence with classes in $H^2(Y;\mathbb{Z})$. 
...
A: In a sense I think things are usually reversed. 
You might find it helpful to read the basics of obstruction theory, for example out of Whitehead's book (but Milnor and Stasheff is good, too).  It indicates a general phenomenon (a thread of ideas engulfing both Serre's dissertation, Thom's work on cobordism and many other problems) where one is interested in determining if two spaces are homotopy-equivalent, or if two maps are homotopic.  One then reduces that homotopy problem to a sequence of homological problems. 
This perspective sees homology as the "small computable increment" towards understanding something about homotopy. 
A: Higher homotopy groups arise in the study of topological defects in physics.
Many physical systems are described by an "order parameter", a manifold valued field on a region of space, $f:U\subset \mathbb{R}^3\rightarrow M$. For example, in some materials there is a continuous (if you don't look too closely) vector valued magnetization property defined at each point in space. The vector has a constant length fixed by the nature of the material. So we have a field $f:\mathbb{R}^3\rightarrow S^2$. In a certain type of liquid crystal the molecules are shaped roughly like line segments so at each point the orientation of the molecules is given by a line. So we have a field $f:\mathbb{R}^3\rightarrow \mathbb{R}P^2$. Dislocations in crystals give rise to maps $f:\mathbb{R}^3\rightarrow T^3$ where $T^3$ is the 3-torus.
Sometimes these order parameters have singularities, points or curves where $f$ is no longer defined. These can be studied by looking at the region around the "defect". For example, if $f$ has a point singularity at the origin then we can look at $f$ on a sphere, $S^2$, around the defect. This gives rise to the restriction $f':S^2\rightarrow M$ which gives an element of $\pi_2(M)$. If $f$ is continuous everywhere except at the origin, then continuously moving the sphere outwards from the origin won't change which homotopy class $f'$ represents. So if $f'$ represents a non-trivial homotopy class then in some sense the defect can't be hidden or deformed away and can be detected on the surface of a sphere a long way from the defect itself. These phenomena have a variety of physical consequences.
(Defects along a line give rise to elements of $\pi_1(M)$ and elements of $\pi_3(M)$ can give rise to a non-localisable "texture defect".)
There is a substantial amount of literature on these defects and they apply to everything from solid and liquid crystals to ferromagnets and cosmic strings.
There is a wikipedia article on topological defects. Here is a paper with some pictures of point defects in liquid crystals.
Higher homotopy groups also arise in the (hot right now) field of topological insulators.
A: If you want to build spaces inductively, then you need to know what ways you can attach an $n$-ball to a given space.  Since the boundary of that $n$-ball is an $(n-1)$ sphere you need to know $\pi_{n-1}$.  So you want to think about homotopy groups just to make examples (or to describe your known examples concretely) even before you start asking questions about those spaces.
A: Here is a simple example: even if all you care about is computing $\pi_1$, sometimes the easiest way to do it is to use the long exact sequence of a fibration, which requires knowing something about a $\pi_2$ at least. For example, the standard (as far as I know) way to compute $\pi_1(\text{SO}(n)), n \ge 3$ (an important computation as it tells you that the spin groups exist!) is to use the long exact sequence of the fibrations
$$\text{SO}(n) \to \text{SO}(n+1) \to S^n$$
the relevant part of which goes 
$$\cdots \to \pi_2(S^n) \to \pi_1(\text{SO}(n)) \to \pi_1(\text{SO}(n+1)) \to \pi_1(S^n) \to \cdots$$
and once you know that $\pi_1(S^n)$ and $\pi_2(S^n)$ are both trivial you can conclude that for $n \ge 3$ we have isomorphisms $\pi_1(\text{SO}(n)) \to \pi_1(\text{SO}(n+1))$. This reduces the computation for all $n$ to the computation for $n = 3$, where $\text{SO}(3) \cong \mathbb{RP}^3$ has universal cover $\text{SU}(2) \cong S^3$. Similar fibrations can be used to compute the fundamental groups of various other classical Lie groups and related spaces. 
You can try to do this computation by computing $H_1$ instead and using Hurewicz but I expect this would actually be more annoying; if you wanted to use the fibrations above then instead of the long exact sequence you'd need to use the Serre spectral sequence. 
A: There is Whitehead's theorem: if $X$ and $Y$ are connected CW complexes and $f: X \to Y$ is a map inducing an isomorphism on $\pi_k(-)$ for all $k \geq 1$, then $f$ is a homotopy equivalence.
A: Simply-connected 4-manifolds are classified by data including the second homotopy group (in the guise of the second homology group). In case you complain that one might as well use $H_2$, certain non-simply-connected 4-manifolds are classified by $\pi_1$, $\pi_2$ as a $\pi_1$-module and some extra data involving $\pi_2$.
(apologies for brief and sketchy answer!)
