This is a soft question to understand better why differential forms are so useful, especially in applied mathematics.
I own Tu's - Introduction to Manifolds, and I'll quote the introduction of section 18:
Differential forms are generalizations of real-valued functions on a manifold. Instead of assigning to each point of the manifold a number, a differential k-form assigns to each point a k-covector on its tangent space. For k = 0 and 1, differential k-forms are functions and covector fields respectively Differential forms play a crucial role in manifold theory. First and foremost, they are intrinsic objects associated to any manifold, and so can be used to construct diffeomorphism invariants of a manifold. In contrast to vector fields, which are also intrinsic to a manifold, differential forms have a far richer algebraic structure. Due to the existence of the wedge product, a grading, and the exterior derivative, the set of smooth forms on a manifold is both a graded algebra and a differential complex. Such an algebraic structure is called a differential graded algebra. Moreover, the differential complex of smooth forms on a manifold can be pulled back under a smooth map, making the complex into a contravariant functor called the de Rham complex of the manifold. We will eventually construct the de Rham cohomology of a manifold from the de Rham complex. Because integration of functions on a Euclidean space depends on a choice of coordinates and is not invariant under a change of coordinates, it is not possible to integrate functions on a manifold. The highest possible degree of a differential form is the dimension of the manifold. Among differential forms, those of top degree turn out to transform correctly under a change of coordinates and are precisely the objects that can be integrated. The theory of integration on a manifold would not be possible without differential forms.
From this introduction, and plus what the text actually covers on the subject, it seems to me that the main advantage of the differential forms over vector fields is their algebraic structure (which is much richer than the vector fields one).
However I do struggle to understand if once I get used to the formalism it actually makes easier to model certain geometric problems. There's another book from the same author (Tu's - Differential geometry) where the formalism of forms is used to derive the structure equations, which are still differential forms.
I own such book as well, but I also own Do Carmo's - Riemannian Geometry, and I've been reading the three of them lately.
The thing is Do Carmo doesn't seem to make use of differential forms at all to develop his theory (except in the exercises), Tu instead uses differential forms as I mentioned.
This difference strikes me, since I can't quite figure what sort of "modelling" advantage differential forms give over normal vector fields to model certain problems.
There're also few papers (Computer Graphics papers) that I've been reading through lately where the differential forms formalism is actually used to design certain algorithms to process meshes, so they must have some modelling advantage that I don't understand.
Can anyone clarify?
Update: As further clarification I think I can give, I'm asking from the applied mathematics point of view. For example in triangular meshes processing, tools from differential geometry are used to compute laplacian, gradient etc, but the methods I'm aware don't really use differential forms, but I'm aware from some papers I've been reading in the last few months that differential forms are somehow used to model some problems, I can't understand though why such formalism is actually better in some situations.
Please let me know if you like some quotes from some of the paper I'm talking about.