We know that there is an equivalence of categories between the two following categories:
$1)$ Classical varieties over $k$, where $k$ is an algebraically closed field. (Informally I mean locally ringed spaces formed by patching affine irriducible algebraic sets over $k$. This is the definition of algebraic variety present for example in Perrin's book.)
$2)$ Schemes over $k$ which are integral and of finite type over $k$
Now I need an example, if it does exist, of two non isomorphic (as classical varieties) algebraic sets that are non isomorphic as $k$-schemes (this is obvious by the previous functor) but isomorphic in the category of schemes.
To be more specific I need two non isomorphic algebraic sets such that when I use the above functor $1)\longrightarrow 2)$ and then I forget the structure morphism on Spec $k$, I finally obtain two isomorphic schemes.
Thanks in advance