Why differential forms are important? Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated without them: for example if you want to learn the definition of symplectic manifold, you must first learn what is differential form. 
However I have heard another opinion from people with non-geometric background (say, functional analysis, probability) that differential forms are probably used only in problems where they already appear in basic definitions (like symplectic geometry). When I taught an undergraduate course "Analysis on manifold" I had a feeling that my students might have got the same impression at the end: in the second half of the semester I developed calculus of differential forms with rather few concrete applications.  Personally I am completely convinced in usefulness of differential forms, but find it not so easy to explain this to people who are not used to them.
I would like to make a list of concrete applications of differential forms. Also it would be interesting to know historical reasons to introduce them. To start the list, below are some examples which come to my mind. 
1) The general Stokes formula $\int_M d\omega=\pm \int_{\partial M}\omega$ generalizes all classical formulas of Green, Gauss, Stokes.
2) The Haar measure on a Lie group can be easily constructed as a translation invariant top differential form. (Some experts often prefer to consider this as a special case of a more general and more difficult fact of existence of Haar measure on locally compact groups. In that generality there is no such a simple and relatively explicit description of Haar measure as there is no language similar to differential forms.)
3) The cohomology of the de Rham complex of differential forms on a manifold is canonically isomorphic to singular cohomology (with real coefficients) of the manifold. One use of this isomorphism is that even Betti numbers of a symplectic manifold are non-zero. Another non-trivial use of this fact is the Hodge decomposition of the cohomology (with complex coefficients) of a compact Kahler manifold which makes it, in particular, a bi-graded algebra (rather than just graded) and provides new information on the Betti numbers, say $\beta_{2k+1}$ must be even.
 A: Differential forms are used in geometric measure theory to define currents.  One use of currents is as a generalization of submanifolds, with better compactness properties.  That is, it's easier to show that a subsequence of currents converge to a current.  This can give one approach to the minimal surface problem; see the encyclopediaofmath.org article on currents.
Maybe someone else can give some better detail or more uses of currents.
A: Differential forms and exterior derivatives are used in Cartan's moving frame method, which allows one to calculate the curvature and Levi-Civita connection of a Riemannian manifold quite elegantly.
Normally the differential form calculations are shorter and less error prone than if one where to try to use Christoffel symbols.  
The method is especially well adapted to cases where it's easy to find a section of the orthonormal frame bundle.  Examples include: Conformal metrics (say the Poincaré metric), Thurston's NIL and SOL geometries on $\mathbb{R}^3$, and, more generally, invariant metrics on Lie groups.
A: There is at least one more important example where people who believe that "differential forms are used only in problems where they already appear in basic definitions" are completely wrong. Differential forms and the exterior differential were created (or discovered) by Elie Cartan in the 1890's in order to solve the Pfaff problem, as follows.
Let us think in $R^n$. Certainly these people will admit that a differential form of degree one, that is, a covector field $$\omega=f_1dx_1+\dotsb+f_ndx_n$$ is a natural object and deserves some attention…. Here $f_1, \dotsc, f_n$ are real functions of $x_1, \dotsc, x_n$. A very natural and universal method in sciences, in order to study an object, is to consider it in the coordinates system where it gets the simplest form…. Pfaff's problem is: given such a form $\omega$, in a small neighborhood of a given point, can I reduce the number of variables to some $k<n$? In other words, can I find local curvilinear coordinates $y_1, \dotsc, y_n$ such that $$\omega=g_1dy_1+\dotsb+g_kdy_k$$ where $g_1, \dotsc, g_k$ are real functions of $y_1, \dotsc, y_k$?
In order to solve this, Cartan defines the exterior differential $2$-form $d\omega$, and considers what he calls the successive differentials of $\omega$, namely:
\begin{align*}
& \omega'=d\omega, \\
& \omega''=\omega\wedge d\omega, \\
& \omega'''=d\omega\wedge d\omega, \dotsc.
\end{align*}
Generally, he defines $\omega^{(k)}$ as $(d\omega)^{(k+1)/2}$ (using the wedge product) if $k$ is odd, and as $\omega\wedge(d\omega)^{k/2}$ if $k$ is even. He proves that the exterior differential $d$ is invariant under coordinates changes (quite a discovery!) and concludes that a necessary condition for the $k$-reducibility is that $\omega^{(\ell)}$ vanishes for every $\ell>k$. Then, his real work is to prove that this necessary condition is sufficient, in the real-analytic case (the smooth case requires some more subtle nondegeneracy condition, if I'm correct).
Also note how these simple formulas contain a lot of geometry. Foliations, symplectic and contact geometries, appear here in the consideration of the (non)vanishing of the successive derivates. Differential forms were indeed one of Elie Cartan's great discoveries.
A: The first sections of Whitney's book on Geometric Integration Theory contain an attempt to motivate the differential forms formalism in low dimensions.
A: Implicit but rarely stated above is the whole range of cohomological physics arising from the differential forms appearing in physics so often.
A: The Maxwell's equations become infinitely more elegant (not to mention explicitly covariant) when stated in the differential form language.
A: When I think of differential forms, two remarks from lecturers stick out:


*

*“The definition of differential form is: something you can integrate.” (from Joe Harris)

*“If you think that functions can be integrated in a natural way, then I ask you: What is the integral of the constant function 1 over this blackboard?” (from Alexander Kirillov) 


We are using differential forms all the time: when integrating a function over an interval, we are tacitly multiplying that function by $dx$.  When computing the flux of a vector field $P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}$ over a surface, we are integrating $P \,dy\wedge dz + Q \,dz\wedge dx + R\, dx\wedge dy$.  In a Riemannian manifold the metric determines a volume form, but in local coordinates that form is still a multiple of $dx_1 \wedge \dots \wedge dx_n$.
So I would answer the question with: Differential forms are important because they are a natural way to express and generalize the idea of integration.
A: In Chern-Weil Theory, characteristic classes appear as certain closed differential forms associated to vector bundles with connections (constructed via the curvature form of the connection). 
When you deal with Clifford bundles, e.g. spinors (which are an interesting thing to study from the physics perspective, I hear), there is Getzler's symbol map that sends Clifford elements to differential forms; in this sense, the Clifford Algebra appears as a "quantized version" of the Grassmann Algebra.
When you follow the Heat Kernel Proof of the Atiyah-Singer Index Theorem, differential forms jump out of the semiclassical asymptotics of the heat kernel using Getzler's symbol mapping, and by some miracle, these are exactly the $\hat{A}$ genus and the Chern character, defined by Chern-Weil theory.
A: Recall that the determinant of a matrix $A$ is the volume of the parallelepiped formed using the columns of that matrix. If I have an $n$ manifold and the  differential $n$-form $\omega = dx^1 \wedge \dotsb \wedge dx^n$ and feed it $n$ vectors then this gives me $$\omega(v_1,\dotsc,v_n) = \det( dx^i(v_j) ).$$ This is precisely the volume of the $n$-parallelepiped formed using the $v_i$'s as columns.
Now for a $k$-form on an $n$-manifold, it is a similar idea. If $\eta$ is a $k$-form and we hit it with $k$ vectors, it should give us the volume of a $k$-parallelepiped (the parallelepiped should live in the plane spanned by the vectors fed into the $k$-form). These can be used to determine the the area of $k$ submanifolds. The usual thing we do if we have a submanifold is to use the imbedding map to pull back the volume form to get a $k$-form on the submanifold.
The point of all this is it naturally gives us integration over manifolds. The vectors we choose to feed into the differential form come from the tangent space of the manifold/submanifold. We can gridify the manifold using the homeomorphism and get tangent vectors using the homeomorphism (for smooth manifolds at least) and use the $k$-form we chose to set up integrals and get volumes.
A: From a more topological viewpoint, I think it is worth reading the famous paper of Dennis Sullivan, Infinitesimal computations in topology (Publications Mathématiques de l'IHES, 1977), in which he builds algebraic models of manifolds from their differential forms. He explains how to construct recursively such models and gives several examples. He obtains from this construction deep results about homotopy types and diffeomorphism types of manifolds.
