The affine line and its punctured principal open are birational but not isomorphic

Birational affine varieties need not be isomorphic.

Assume the Axiom of Choice and let $k$ be an algebraically closed field. Take $X={\mathbf{A}}_k^1$ with coordinate ring $k[t]$ and $U=D_X(t)={\mathbf{A}}_k^1\setminus \{0\}\mathrm{.}$ The explicit isomorphism in The hyperbola xy = 1 is isomorphic to the punctured affine line identifies $U$ with the classical affine variety $V(xy-1)\subseteq {\mathbf{A}}_k^2$, so $U$ may be used as an affine-variety target. The identity on the common nonempty open $U$, in one direction, and the inclusion $U\hookrightarrow X$, in the other, exhibit $X$ and $U$ as birational.

They are not isomorphic. By Regular functions on a principal open are the principal localization of the coordinate ring, $k[U]\cong k[t,t^{-1}],$ whereas $k[X]=k[t]$. Every unit of $k[t]$ is constant: if $fg=1$ in $k[t]$, then $\mathrm{deg}(f)+\mathrm{deg}(g)=0$, so $\mathrm{deg}(f)=0$. By contrast, $t$ is a nonconstant unit of $k[t,t^{-1}]$ with inverse $t^{-1}$. Therefore the two rings have different unit groups and cannot be isomorphic. Thus Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms rules out an isomorphism of varieties.