Suppose $X$ and $Y$ are smooth affine surfaces over $\mathbb C$. Suppose there is a biholomorphism $f: X\to Y$. Does it follow that $X$ and $Y$ are isomorphic as affine surfaces (i.e. there exists an algebraic isomorphism $g: X\to Y$)?
What if we additionally know that $X$ and $Y$ are rational surfaces?
Let $\overline{C}$ be a complex projective curve of genus $g>0$. Let $C\subset \overline{C}$ be the open affine complement of one (closed) point $p$. The composition $\text{Pic}^0(\overline{C})\to \text{Pic}(\overline{C})\to \text{Pic}(C)$ is an isomorphism. Let $L$ be any nontrivial (geometric) rank $1$ vector bundle over $C$. This is affine, since $C$ is affine and the projection morphism from $L$ to $C$ is affine. This projection morphism equals the image of the Albanese morphism of $L$ (this is a birational invariant for complex projective manifolds, thus extends unambiguously to complex quasi-projective manifolds). The relative tangent bundle of this projection morphism is the pullback from $C$ of a unique invertible sheaf, namely the invertible sheaf associated to $L$. Thus, the relative tangent bundle of the Albanese morphism is not trivial. Hence $L$ is not isomorphic to $\mathbb{A}^1\times C$. Yet the underlying complex manifolds are biholomorphic.
Update: Note that this fails to provide the required example since the second algebraic structure is not affine as desired. However, I am not deleting this answer as yet.
The moduli of flat line bundles over a curve $X$ of genus $g$ is parametrized holomorphically by $\left(\mathbb{C}^*\right)^{2g}$. This uses the topological description of such a line bundle in terms trivialisation on the universal cover. This yields one algebraic structure given by $(\mathbb{G}_m)^{2g}$. In particular, this has no morphisms to an Abelian variety.
On the other hand, a line bundle with a flat connection has degree 0. Different flat connections on a line bundle differ by a holomorphic $1$-form on $X$. This shows that the algebraic moduli space of flat connections on a line bundle over $X$ is an affine space torsor $T$ for $H^0(X,\Omega^1_X)$ over $\mathrm{Pic}^0(X)$. In other words, $T\to \mathrm{Pic}^0(X)$ is a bundle of $\mathbb{A}^g$'s.
Thus $T$ and $(\mathbb{G}_m)^{2g}$ are two different algebraic structures (since one maps to $\mathrm{Pic}^0(X)$ and the other does not) for the same underlying holomorphic manifold of dimension $2g$. The original question (for surfaces) is answered by taking $g=1$.
@aglearner has asked how we see that there is an algebraic moduli space for flat connections of a fixed dimension $r$ over a smooth projective variety $X$. The most comprehensive reference is the work of Simpson here and here. However, there ought to be easier to read explanations when $X$ is a curve and $r$ is 1, but I could not find them!
