So I've seen these defined when constructing differential forms and in the construction of integration of manifolds. However, these seem to be a standard subject in most graduate algebra books, yet, I've never seen them applied anywhere else than in differential geometry.
My question is: What use do these have as abstract objects? What information do they convey?
The exterior powers induce operations on (algebraic or topological) K-theory. Given an f.g. $R$-module $P$ (or a vector bundle over some space), the $n$th exterior power module $\Lambda^{n}P$ is again projective, and so induces a self map of the Grothendieck group $\lambda^{n} : K_{0}(R) \rightarrow K_{0}(R)$. If we do so for each $n$ we give $K_{0}(R)$ the structure of a $\lambda$-ring. Taking exterior powers of vector bundles over a fixed space $X$ puts a $\lambda$-ring structure on topological K-theory $K^{0}(X)$ in exactly the same way.
Note that the $\lambda$-operations are not homomorphisms, but given a $\lambda$-ring structure we may formally define Adams operations on that ring in a canonical way. The Adams operations are ring homomorphisms, so they can be used to deduce further information about $K^{0}(X)$. Adams used this method to solve the Hopf invariant 1 problem, which resolves the question of which spheres are $H$-spaces (as a corollary of this we can conclude that the only finite dimensional normed division algebras over $\mathbb{R}$ are the reals themselves, the complex numbers, the quaternions and the octonions).
In the same way as mentioned above, the symmetric power functors also induce operations on K-theory, often denoted $\sigma^{n}$. Both the exterior powers and the symmetric powers are special cases of Schur functors. As well as the applications in K-theory, Schur functors have applications in the representation theory of symmetric groups.
Symmetric algebras are polynomial algebras. Polynomials appear all over the place. Exterior algebras also appear all over the place. They can be thought of as graded polynomial algebras.
One simple example is that if $\mathfrak g$ is a Lie algebra, then $$\cdots\to\Lambda^{k+1} \mathfrak g \to \Lambda^{k} \mathfrak g\to\cdots$$ is a chain complex, where the boundary operator is defined by $$\partial(x_1\wedge\cdots\wedge x_k)=\sum_{i<j}(-1)^{i+j+1}[x_i,x_j]\wedge x_1\wedge\cdots\wedge \hat{x_i}\wedge\cdots\wedge \hat{x_j}\wedge\cdots\wedge x_k.$$ Here the "hat" notation means to remove those terms from the long wedge. Lie algebra homology is $H_*(\mathfrak g)$ is defined as the the homology of this complex. (This is called the Chevalley-Eilenberg complex.)
The fact that $[\cdot,\cdot]$ satisfies the Jacobi identity is equivalent to $\partial^2=0,$ and this the key to defining a more flexible gadget callet an $L_{\infty}$-algebra, which is any coderivation on $\Lambda\mathfrak g$ satisfying $d^2=0$.
