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 $H\_k$?



Homotopy groups other than $\pi_1$ : what are they good for?
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. 


