I know Riemann surfaces are actually algebraic curves, i.e. all Riemann surfaces can be simply embedded into some projective space $\mathbb{P}^n$. But this doesn't indicate me more correspondences between analytic functions and algebraic functions. (For example, holomorphic 1-forms and regular 1-forms.)
On the other hand, I learn such correspondences from Serre's GAGA. This is valid for projective varieties of any dimensions, but were there such theorems for Riemann surfaces (before GAGA)? Or, GAGA had an impact for just the case of Riemann surfaces?
1) Contrary to a widespread misconception, GAGA in no way proves that a compact Riemann surface has an algebraic structure: GAGA starts with the datum of an algebraic variety.
Riemann's theorem according to which a compact Riemann surface $X$ can be embedded into $\mathbb P^3(\mathbb C)$ still requires some difficult results from analysis.
Gunning's lecture notes were the first didactical exposition to give a modern proof relying on methods from sheaf cohomology:
Modern proofs rely on the finite dimensionality of the complex vector space $H^1(X,\mathcal O_X)$ (a special case of a result by Cartan-Serre), and that in turn uses a functional analysis result due to Laurent Schwartz on the perturbation of a surjective morphism between Fréchet spaces by a compact morphism.
For a masterful and detailed proof of the embedding theorem Forster's book is, as usual, the best reference.
2) However GAGA implies that up to isomorphism the algebraic structure on a compact Riemann structure is unique (up to isomorphism).
This follows from the GAGA result that any holomorphic map between projective complex algebraic varieties is automatically regular.
Notice that the unicity of an algebraic variety structure with given underlying holomorphic manifold cannot be expected in the non compact case: see here.
3) Once one knows that a compact Riemann surface $X$ is automatically algebraic, GAGA has very interesting consequences.
The most elementary is that any meromorphic function on $X$ is in fact rational: a very classical, pre-GAGA theorem.
A more modern result is that every holomorphic vector bundle on $X$ has a unique algebraic structure and that the corresponding cohomology vector spaces calculated in the classical topology and the Zariski topology are canonically isomorphic.
In particular this applies to the line bundle $\Omega^1_X$ of $1$-forms and taking sections of that bundle (= zeroth cohomology vector space !) we see that every holomorphic $1$-form on $X$ is in fact regular, as evoked by Tei in his question.
4) As @guy-in-seoul very pertinently comments, the full force of GAGA for cohomology of vector bundles (or coherent sheaves) could not have been known pre-GAGA since the very concept of cohomology in the Zariski topology was not known before Serre introduced that notion in his essentially simultaneous FAC.
Notice also that the 1950's were a very exciting time for cohomology in algebraic geometry: there would be much to say on the relations between GAGA, Serre duality (him again!), Dolbeault cohomology, Hodge theory,...