I've decided to expand my comment into an answer.

The point is what do you think a topology is for. If you think that a topology is for talking about convergence of sequences, then no the Zariski topology doesn't do anything for you in that regard. However I contend that in geometry (even differential geometry) people don't use a topology because of convergence issues, but to speak of local properties. That is you may want to say e.g. that a function is 0 not only at a point, but also in a neighbourood of that point. Or you may want to say that you can define an object locally and then try to patch it up to a global object. The Zariski topology gives you exactly this.

This is what I mean, loosely speaking, when I say that the Zariski topology is there to do sheaf theory. One could argue that the whole purpose of sheaf theory is to bridge the gap between local and global properties. If your only experience so far comes through a differentiable setting (for instance multivariable calculus) you may not appreciate how hard these questions are in general. This is because in a differentiable setting you have partitions of unity, which roughly speaking, tell you that the sheaf theory is trivial. However in an algebraic (or complex analytic) setting you do not have that luxury anymore.

As a final note, working in algebraic geometry people noticed that they don't really need a topology on the underlying set to do sheaf theory (that is, to speak about local properties), but that there is a more abstract notion of a Grothendieck topology that does the trick. So when people speak about the "étale topology" or the "fppf topology" as a refinement of the Zariski topology, they don't mean a genuine topological space but one of these more abstract generalizations.

I've decided to expand my comment into an answer.

The point is what do you think a topology is for. If you think that a topology is for talking about convergence of sequences, then no the Zariski topology doesn't do anything for you in that regard. However I contend that in geometry (even differential geometry) people don't use a topology because of convergence issues, but to speak of local properties. That is you may want to say e.g. that a function is 0 not only at a point, but also in a neighbourood of that point. Or you may want to say that you can define an object locally and then try to patch it up to a global object. The Zariski topology gives you exactly this.

This is what I mean, loosely speaking, when I say that the Zariski topology is there to do sheaf theory. One could argue that the whole purpose of sheaf theory is to bridge the gap between local and global properties. If your only experience so far comes through a differentiable setting (for instance multivariable calculus) you may not appreciate how hard these questions are in general. This is because in a differentiable setting you have partitions of unity, which roughly speaking, tell you that the sheaf theory is trivial. However in an algebraic (or complex analytic) setting you do not have that luxury anymore.

As a final note, working in algebraic geometry people noticed that they don't really need a topology on the underlying set to do sheaf theory (that is, to speak about local properties), but that there is a more abstract notion of a Grothendieck topology that does the trick. So when people speak about the "étale topology" or the "fppf topology" as a refinement of the Zariski topology, they don't mean a genuine topological space but one of these more abstract generalizations.

EDIT: as requested in the comment, let me put here a proof that the cohomology of $C^\infty$-modules is trivial. To be precise what I'll prove is the following: if $0\to \mathcal{F}'\to \mathcal{F}\to \mathcal{F}''\to 0$ is a short exact sequence of sheaves of $C^\infty$ modules then for every open set $U$ the sequence $$ 0\to \mathcal{F}'(U) \to \mathcal{F}(U)\to \mathcal{F}''(U)\to 0$$ is exact. In fact, by general results, the only thing needed to prove is the surjection of the last map. Take a section $s''\in \mathcal{F}''(U)$. Since the sequence of sheaves is exact we may find a cover $\{U_i\}_i$ of $U$ and sections $s_i\in \mathcal{F}(U_i)$ such that the image of $s_i$ in $\mathcal{F}''(U_i)$ is $s|_{U_i}$. Let now $\psi_i\in C^\infty(U)$ a partition of unity associated with the cover. I claim that the expression $$ s = \sum_i \psi_i s_i$$ makes sense as an element of $\mathcal{F}(U)$. In fact this expression makes sense locally around every point of $U$ (since locally only a finite number of $\psi_i$ are nonzero) and the sheaf property says that these patch to give us a global section. Then it is easy to check that the image of $s$ in $\mathcal{F}''(U)$ coincides with $s''$ locally around any point, and so by the sheaf property, must coincide globally.