For example, consider the differential one-form $$\frac{\mathrm dw}{1-w^2}$$ If we make the change of coordinates $w=1/z$ then we see that $$\frac{\mathrm dw}{1-w^2} \longrightarrow \frac{\mathrm dz}{1-z^2}$$ Is there any significance to the form "being the same" in both coordinate systems?
Context: I'm looking at meromorphic differential forms on Riemann Surfaces.
On
We can think of the substitution $w\leftrightarrow w^{-1}$ as an action of the cyclic group of order $2$ on the Riemann sphere. The quotient by this action is again the Riemann sphere, and we can identify the quotient map with $$ \begin{align} \mathbb{CP}^1&\to\mathbb{CP}^1\\ w&\mapsto t:=w+w^{-1}.\end{align}$$ The property that the one-form $\omega:=dw/(1-w^2)$ is invariant under $w\leftrightarrow w^{-1}$ means that $\omega$ actually comes from a one-form on the quotient. In other words, $\omega$ can be written in terms of $t$: $$ \frac{dw}{1-w^2}=\frac{dt}{4-t^2}. $$
Then you can define a global form by defining in each coordinate chart and patch them together.