Well for me, for these simple shapes, the advantages of the path were it being faster and more convenient.
Maybe I need to improve my bezier skills. But for example, to create a spiral shape with a bezier, after extruding each control point I would need to rotate and scale each handle to get a smooth path.
With a path, I can do this faster by simply clicking and dragging the points. Both the general spiral shape, and a smoother curvature between the points, are achieved faster and with less tweaking by using a path for me.
Another thing is when using the radius slider to thicken a part of the curve. With bezier, by default, it gives a sharp edge (it is set to linear). We can go to the object properties, then to the active spline section, and change the radius interpolation to Bspline. Which for creating these pieces is an extra step not required when using a path.
But of course like I said in the video, there are different ways of modeling the same thing. If anyone would have done this differently, please do share your workflow!