A differentiable form on an open subset $U \subseteq\mathbb{R}^2$ is a map $$\omega=\omega_xdx+\omega_ydy$$ for some $\omega_x,\omega_y \in C^1(U)$.
For example, if $f \in C^1(U)$, $\omega=df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$ is a differential form.
My question is: what is the difference between $\partial x$ and $dx$?
On
There may be no such things as $\partial x$. You have the operator $\frac{\partial}{\partial x}$. It is a member of the tangent space. As such, it is a differential operator on the space of germs of fucntions around a point.
$dx$ is in the co-tangent space. It is a function on the tangent space.
You have $dx \left (\frac{\partial}{\partial x} \right ) = 1$ by definition
$\mathrm{d}x$ is a basis vector in the space of 1-forms for functions in $C^1$. On the other hand, $\partial{x}$ is not a symbol with any meaning: it does denote any particular mathematical object. The symbol $\partial{x}$ is merely an artifact of notation: $\frac{\partial{f}}{\partial{x}}$ is merely how we choose to denote the partial derivative of $f$ with respect to $x$. We could choose to denote it differently, avoiding the symbol $\partial{x}$ altogether, since it has no mathematical significance. Why was this symbol chosen? I am sure there is some history behind the symbol, though I am not knowledgeable in it. But one could very well choose to denote it as $f_x$ instead, and now you communicate the same idea, without appealing to strange notation.