What's the advantage that polar coordinates have over Cartesian coordinates?

Polar coordinates can often describe a situation with circular symmetry more conveniently than Cartesian coordinates. For example, for a ball moving in a circle on a string, only $\phi$ changes, whereas you'll have more complicated expressions in $x$ and $y$.

Similarly, as we'll see later in the course in many examples, if a situation has cylindrical symmetry in 3D, cylindrical coordinates often work pretty well, and spherical coordinates work well for spherical symmetry.