I know from Hartshorne (II.5.10), they are two different closed subschemes in the affine plane Spec $k[x,y]$. But it seems to me they have the same underlying topological space, which is the curve $y=x^2$ and the generic point. So I guess they should have different structure sheaves separately ($\mathcal{O}_{\text{Spec } k[x,y]/(y-x^2)}$ and $\mathcal{O}_{\text{Spec } k[x,y]/(y-x^2)^2}$).
Am I right? and how to see this result better? (how those two sheaves are different...) Thank you!!