Since you asked for "comments, suggestions" here are two thoughts off the top of my head:
- In Torsion Homologique et Sections Rationnelles by Grothendieck (Séminaire Chevalley, 1958), the case for principal bundles is worked out. In particular, for $G$-principal bundles over $\mathbb{C}$ where $G$ is isomorphic to a product of groups of type $\operatorname{SL}_n$ or $\operatorname{Sp}_n$, if the (algebraic) bundle is locally trivial in the analytic topology then it will be locally trivial in the Zariski topology.
- To learn how to "work around" the difference for cohomological computations, see the propositions in Section 2 in Hodge polynomials of $\operatorname{SL}(2,\mathbb{C})$-character varieties for curves of small genus by Logares, Muñoz, and Newstead.
