It’s fair to say that a polygon is a simple and flexible building block.
However, a parametric representation of an object has certain key advantages:
Conciseness
A parametric representation is exact and economic since it is analytical.
With a polygonal object, exactness can only be approximated at the expense of
extra processing and database costs.
Deformation and shape change
Deformations of parametric surfaces is no less well defined than its undeformed
counterpart, so the deformations appear smooth. This is not generally the case
with a polygonal object.