The "PDE" approach is working by effectively setting some boundary conditions and solving a differential equation. The smoothness of the contours comes from the fact that they are generated implicity by a relativity simple set of parameters. A typical vector format would instead record each contour explicitly, which, even after smoothing, is a lot of data.

The big idea is to describe the image intensities as a continuous surface which can be approximated in 3+ dimensions, not a patchwork of vector areas.