In my complex analysis class, I learned that a complex function $f(z)$ is said to be analytic at $z_0$ if there is a neighborhood around $z_0$ in which $f$ is differentiable. Then, I learned various properties of analytic functions, but I never quite understood why the notion of pointwise differentiability is not enough and we in fact need differentiability "in nearby neighborhoods".
What is the importance/role of "neighborhoods" in this context? What goes wrong if one replaces analyticity with differentiability? I thought that when one talks about pointwise differentiability, one would like to work in open sets and since analyticity and differentiablity are the same in open sets, I do not see what is important about defining analyticity the way it is defined
Because while the definition "differentiable in a neighborhood" is a suitably-equivalent way of saying "analytic" in the context of complex analysis, it is not necessarily the most informative way.
The most informative definition of analytic is just the extension of the one from real analysis: a complex $f(z)$ is analytic at $z_0$ iff its Taylor series expansion at $z_0$ converges to $f(z)$ in a neighborhood of $z_0$.
In complex analysis, this is equivalent to the statement that it be differentiable in a neighborhood. In real analysis, it is not equivalent.
In fact, in complex analysis the following equivalences hold: all (complex-)differentiable functions are (complex-)smooth and all smooth functions are analytic. But the definitions of these terms need not be the same, even though we could do so because of these equivalences. IMO, it's better to use the regular definitions and then prove the equivalence as a useful theorem. To go from Real to Complex with a change of definition of "analytic" from "converging Taylor series" to "differentiable in a neighborhood" would be rather jarring, and leave one scratching one's head for reasons, imo. And it obscures some of the beauty, which is precisely that in the complex plane the added structure is far more restrictive on the behavior of the function than in real-number analysis and the reasons for it being so (essentially, that a function which is locally complex-linear (and non-constant) preserves the shape of vanishing objects at (almost) every point, while more general 2D real $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ functions do not.).