What is an analytic function and how is it different from a differentiable complex function?

In complex analysis, the terms "analytic" and "holomorphic" are often used interchangeably, but I want to understand the nuances.

A function $f: \mathbb{C} \to \mathbb{C}$ is complex-differentiable at $z_0$ if:

exists and is independent of the direction from which $h \to 0$.

This leads to the Cauchy-Riemann equations:

How does complex differentiability imply analyticity (power series expansion)? And why is this so different from real differentiability?