The difference is that the scalars of a vector space form an algebraic field while the scalars of a module form a ring. For us humans (as opposed to “those higher beings that I do not understand (a.k.a. mathematicians)”) this means that the scalars in the vector field can divide each other while the scalars in the spaces spanned by fields (i.e. a module) can not.
And just so we all can become even more confused: This has nothing to do with the fact that the “components” of each form field or vector field in certain basis are functions, i.e. themselves elements of a vector space with infinite number of dimensions.
The first way to see this module-not-a-vector-space characteristic is by showing directly that the scalars that form the “coordinate components” of a vector field can not always be divided, even if they are not identically zero. Take, for instance the, manifold with the polar coordinate system and look at the vector . The “scalars” are and . Obviously we can not divide the former by the latter because it will be undefined at .
Another, more amusing way to show that the space spanned by these fields is not a vector space is to explicitly show that a property expected from vector spaces is not fulfilled. Namely, that in dimensions an -uple of linearly independent elements forms a basis. However, in the case of fields over a manifold we can easily have a number of fields that are linearly independent over the manifold as a whole, and are at the same time linearly dependent (or simply equal to zero) on a subdomain. Hence, we have an -uple of linearly independent fields that can not be linearly combined to represent another arbitrary field.