Algebraic de Rham cohomology vs. analytic de Rham cohomology

If $X$ is smooth and proper, GAGA does in fact suffice (despite the observation that $d$ is not $\mathcal{O}_X$-linear: One obtains a comparison map of hypercohomology spectral sequences; it is an isomorphism on the $E_2$ page by GAGA, and thus on the $E_\infty$ page.

It is to prove the general case (i.e., $X$ smooth but not necessarily proper) that one needs to do additional work.

I don't think you can get this directly from GAGA. The reference that I know for this result is Grothendieck, On the de Rham cohomology of algebraic varieties. It is short, beautiful, and in English.

Various people have answered the question, and also brought up some of the subtleties in applying GAGA. So I won't rehash all that. So let me just suggest the additional reference:

Deligne, Équations différentielles à points singuliers réguliers

especially Chapter II, section 6. These issues are dealt with carefully in a more general setting of de Rham cohomology with coefficients in a regular integrable connection. The result is no doubt true for a regular holonomic D-module, and it would be nice if someone wrote this down carefully. But perhaps I'm straying too far from the original topic.