Let $M$ be a smooth manifold, $\bigwedge^k T^*M$ the subset of covariant $k$-tensors on $M$ that are alternating tensors, and $\Omega^k(M)$ the space of differential $k$-forms.
I have always thought of these two spaces to be the same, but recently I've been convinced that this is not correct. For example, in Lee's book on smooth manifolds he defines $\Omega^k(M)$ as a section of $\bigwedge^k T^*M$: $$\Omega^k(M) = \Gamma\Big(\bigwedge^k T^*M\Big).$$ However it seems that both spaces are exactly the same. For example, an element of $\Omega^k(M)$ assigns to each point $x \in M$ an alternating $k$-linear tensor on $T_xM$. Likewise, an element of $\bigwedge^k T^*M$ does the same at $x$. So why do we define $\Omega^k(M)$ as a section of $\bigwedge^k T^*M$ instead of just $\bigwedge^k T^*M$?
This is not accurate.
The difference between those two structures is essentially the same as between the tangent bundle and the vector fields of a manifold.Loosely speaking, the tangent bundle is the set of all tangent vectors to the manifold (more rigorously, the pairs of points of the manifold and tangent vectors, equipped with a projection map $\pi$ over the manifold of course). Now, a vector field can also be viewed as a set of tangent vectors; but of course, each vector is associated with a certain point of the manifold, and the information of which tangent space the vector belongs to must also be held in mind. The right notion is that of a section: that is, $$\sigma: M\rightarrow TM$$ with the property that $\pi\circ\sigma=id_M$
Likewise, the space of alternating k-tensors $\bigwedge^k T^*M$ functions like the tangent bundle; it's the set of all such tensors. The differential forms are "alternating k-tensor" fields over $M$, so, sections of the alternating k-tensor bundle.