One comment about your sentence "this seems to only trivially invoke representation theory". It might be surprising, but such obvious representations are actually the source of interesting mathematics, and a lot of effort of representation theorists is devoted to studying them.
More precisely: start with a group (in your example $SO(n)$) acting on a space $X$ (in your example $\mathbb R^n$), and look at the space of functions on $X$ (let me write it $\mathcal F(X)$; in a careful treatment, one would have to think about whether we wanted continuous, smooth, $L^2$, or some other kind of functions, but I will suppress that kind of technical consideration).
Then, as you observe, there is a natural representation of $G$ on $\mathcal F(X)$.
You are right that from a certain point of view this seems trivial, because the representation is obvious. Unlike when one first learns rep'n theory of finite groups, where one devotes a lot of effort to constructing reps., in this context, the rep. stares you in the face.
So how can this be interesting?
Well, the representation $\mathcal F(X)$ will almost never be irreducible. How does this representation decompose?
Suddenly we are looking at a hard representation theoretic problem.
First, we have to work out the list of irreps of $G$ (which is much
like what one does in a first course on rep'n theory of finite groups).
Second, we have to figure out how $\mathcal F(X)$ decomposes, which involves representation theory (among other things, you have to develop methods for investigating this sort of question), and also often a lot of analysis (because typically $\mathcal F(X)$ will be infinite dimensional, and may be a Hilbert space, or have some other similar sort of topological vector space structure which should be incorporated into the picture).
I don't think I should say too much more here, but I will just give some illustrative examples:
If $ X = G = S^1$ (the circle group, say thought of as $\mathbb R/\mathbb Z$)
acting on itself by addition, then the solution to the problem of decomposing
$\mathcal F(S^1)$ is the theory of Fourier series. (Note that a function on $S^1$ is the
same as a periodic function on $\mathbb R$.)
If $ X = G = \mathbb R$, with $G$ acting on itself by addition, then
the solution to the above question (how does $\mathcal F(\mathbb R)$ decompose under the
action of $\mathbb R$) is the theory of the Fourier transform.
If $ X = S^2$ and $G = SO(3)$ acting on $X$ via rotations, then decomposing $\mathcal F(S^2)$ into irreducible representations gives the theory of spherical harmonics.
(This is an important example in quantum mechanics; it comes up for example in the theory of the hydrogen atom, when one has a spherical symmetry because the electron orbits the nucleus, which one thinks of as the centre of the sphere.)
If $ X = SL_2(\mathbb R)/SL_2(\mathbb Z)$ (this is the quotient of a Lie group by a discrete subgroup, so is naturally a manifold, in this case of dimension 3),
with $G = SL_2(\mathbb R)$ acting by left multiplication, then the problem of decomposing $\mathcal F(X)$ leads to the theory of modular forms and Maass forms, and is the first example in the more general theory of automorphic forms.
Added: Looking over the other answers, I see that this is an elaboration on AD.'s answer.
I'll begin with Maxwell's Equations. I would say that the shape of Maxwell's equations demand Minkowski geometry where the isometries include Lorentz transformations. Before the advent of relativity, Maxwell's equations were already what we now term "relativistic". So, I'm not sure I agree with your question historically. That said, if we accepted that space and time was prescribed in the Newtonian sense (absolute time, and euclidean stand-alone space) then we have to either discard Maxwell's equations or make bizarre adhoc modifications of classical mechanics. It's an older text, but Resnick is a nice read to see explicitly what the Lorentz transformations are and also to see explicitly how Maxwell's equations are covariant. His proof is explicit at the level of PDEs, which is nice for some students. Others will be more pleased by something like Misner Thorne and Wheeler where the equation is recast in terms of differential forms to break free of the coordinates.
The possible wavefunctions of the hydrogen atom are indicated by it's symmetries. Wavefunctions provide representations of that symmetry group. One interesting idea, if left alone, electrons tend to stay in inequivalent representations of the symmetry group. The Hamiltonian matrix is block-diagonal and no mixing occurs between energy eigen states. If a pertubation occurs then coupling terms appear which join different energy eigen states. From a symmetry perspective, the departure from the perfect symmetry opens new interactions before not possible. I recommend Greiner's text on Lie groups and representation theory in physics. This is certainly not a math book, but it helps answer some of the questions you raise. I found it more readable than other books with similar goals. For gory details on spectroscopy and so forth the classic by Hammermesch still finds a home on the bookshelf of many aspiring theorists who wish to master rep-theory for Atomic and Molecular physics. On the math side of things, the text Symmetry Groups and their Applications by Willard Miller is worth a look.
Spin is a particular number which is associated with matter. It labels how the particle interacts with the magnetic field. Mathematically, it has do with the double-cover of the Lorentz group and the concept of a "spin-bundle" has considerable generalization over the usage of the term in standard physics. Typical examples, electrons have spin 1/2 whereas photons have spin 1. The graviton has spin 2. Spin, like mass or charge of a particle is an intrinsic property. It's part of what defines the identity of a given elementary particle. In short, spin labels the type of spinorial representation of the Lorentz group.
To see how physicists think about it perhaps look at the text by Wu Ki Tung. There is a three-volume set by Cornwell, I suspect there is a reasonable deep explanation in those, but sadly I only own a poor copy of one at the present. Bleeker's Gauge Theory and Variational Principles discusses spin and Lorentz groups, and it's a Dover now. To see a detailed construction of spinors from other algebra there is a text by Naber which appears quite accessible if you have some time. Personally, I haven't worked it out, but it might be of interest. See Geometry of Minkowski Spacetime
Best Answer
Although you specifically asked for applications not from physics, let me begin by mentioning that representation theory is of paramount importance in physics, and once you decide to look for such applications, you will find many! The same goes for chemistry.
Now for applications in pure mathematics. As Tobias mentioned in a comment, two famous applications are the Burnside $pq$-theorem and the structure theorem of Frobenius groups. Both of these are discussed in detail in chapter 6 of my representation theory notes. Isaacs's wonderful book on character theory contains a vast amount of applications of representation theory. For example the classification of finite simple groups is completely unthinkable without representation theory, both classical and modular, and Isaacs gives a glimpse of that. In fact, character theory was invented by Frobenius without representations in mind, and in the attempt to solve a purely group theoretic problem. It was pointed out later by Schur that what Frobenius had really done was representation theory. A sketch of this history is contained in the introduction to the aforementioned notes, but there are better sources.
Representation theory is extremely important in number theory. In particular, there are groups that we don't know how else to begin understanding, other than through their representations, most notably the absolute Galois group of $\mathbb{Q}$. It is very big, and it is not clear how to describe such big groups in a useful way (note that generators and relations are useless for most purposes, since there is not even an algorithm to tell whether a given presentation describes the trivial group). On the other hand, the Galois group by its very nature acts on lots of things, and it is very natural to try understanding it through these actions.
Finally, note that group representations are simply part of our world, so it would be foolish to try avoiding them. In particular, historically one could argue that group representations were born before groups were. This is not true literally, since the definition only appeared in the 20th century, but it is true morally: the first incarnation of groups that people considered was that of symmetries of geometric objects. And those are nothing other than group representations.