As far as I can tell, the topological spaces associated to the schemes in the title are both sets with two elements, with the discrete topology since both have prime ideals $(x)$ and $(x-1)$ which are maximal and thus closed.
In regards to the sheaf structure, obviously the global functions are different. However, as far as I can tell, the local rings of $\mathbb{C}[x]/(x^2-x)$ at $(x)$ and $(x-1)$ are isomorphic to $\mathbb{C}$, as is the local ring of $\mathbb{C}[x]/(x^3-x^2)$ at $(x-1)$, but the local ring of $\mathbb{C}[x]/(x^3-x^2)$ at $(x)$ is a bit larger in some sense (since for example $x$ is not identified with some element of $\mathbb{C}$).
What I would like to know is: how does one translate this algebraic difference (which feels minor to me, because in my mind, both schemes are pairs of points, and if we think about the local rings are local functions, despite the second having more elements in the local ring at $(x)$, there is still only one prime ideal in said local ring to evaluate the functions on, so all functions are still in some sense constant) into a geometrically satisfying picture?