It is said that scheme theory was made to get something that it was not "visible" in classical algebraic geometry, for example as varieties $V(x)=V(x^2)$, in general. If $X$ is an affine variety in $A^n$, for some $n$ natural, and $I\subset K[x_1,..,x_n]$ is an ideal then $V(I)=V(\sqrt{I})$
The varieties are the same, but I read that as schemes they are different. But i don not understand what it means. For this example $V(x)=V(x^2)$, in what sense they are different?
I know that in schemes is not only important the topological space, but the structure sheaf. For example in a topological space $X$, we have the sheaf of continuous functions and the sheaf of differentiable functions, but the same topological space can have many differentiable structures.