Let $M$ be a manifold and let $A = \mathcal{C}^\infty(M)$ be the ring of smooth real-valued functions.
An old posting asks about the relationship of Kähler differentials and ordinary differential forms. While it's clear from the universal properties that there is a surjective $A$-module homomorphism map $\Omega^1_{A/\mathbb{R}} \to T^*(M)$, it is not necessarily an isomorphism. A counterexample for $M = \mathbb{R}$ is given as $\mathrm{d}e^x \neq e^x \mathrm{d}x$ in $\Omega^1_{A /\mathbb{R}}$.
However, it's equally clear, e.g. by this argument, that they they have the same evaluations; if $P$ is a point and we consider the homomorphism $A \to \mathbb{R} : f \mapsto f(P)$, then $$ \mathbb{R} \otimes_A \Omega^1_{A / \mathbb{R}} \cong T_P^*M $$ as vector spaces.
I interpret David Speyer's proof of the above counterexample as considering the evaluations at one of the hyperreal-valued "points at infinity", which has many nonstandard scalars, and for these, $\mathrm{d}r$ is not forced to be zero, so you can make weird things happen
If $M$ is a compact manifold, however, then there are no points at infinity; all of the points are real-valued points. In this case, do we have an isomorphism $\Omega^1_{A / \mathbb{R}} \to T^*(M)$?
