I am looking for some geometric reasoning why we require flatness in the definition of an étale morphism. If we have a morphism $f:X\rightarrow Y$, it seems to me that the condition of being unramified removes the possibility of a fibre “jumping” in dimension however this seems to be the reason for requiring flatness. What problems can the condition of flatness solve that unramified can’t?
For example take the nodal cubic $X=V(y^2-x^3)$ projecting down to $Y=\mathbb{A}^1$. Then this morphism is unramified everywhere except the origin, then would we not also have a flatness away from the origin i.e. if we considered $U=V\backslash (0,0)$ and the morphism $f|_U:U\rightarrow \mathbb{A}^1\backslash \{0\}$ would this not be flat?
Could someone also give an example of an unramified morphism that is not flat?