I am looking for an example of two algebraic vector bundles on an algebraic complex manifold / smooth complex algebraic variety which are analytically isomorphic, but not algebraically isomorphic.
By Serre's GAGA theorem such examples can only exist if the manifold is not projective. I would expect that such examples exist, but I haven't been able to find a reference.
Let $ X $ be the variety which is an elliptic curve $ E $ minus a point $ P $. The analytic line bundles on $ X $ are given by $ H^1(X, \mathcal{O}_X^{\times})$ and this group is isomorphic to $ H^2(X, \mathbb{Z}) $ from the exponential long exact sequence because $ H^1(X, \mathcal{O}_X) = H^2(X, \mathcal{O}_X) = 0 $ by Cartan's Theorem B. So every line bundle on $ X $ is analytically trivial as $ H^2(X, \mathbb{Z})=0 $ - this can be seen by deformation retracting $ X $ to a wedge of two circles, for instance.
However, the algebraic line bundles on $ X $ are given by $ Jac(E) $ by the exact sequence $ \mathbb{Z} \rightarrow Cl(E) \rightarrow Cl(X) \rightarrow 0 $ where the first arrow is $ n \rightarrow n[P] $.