Is the primary advantage of using nonlinear interpolation that we need fewer points to represent the same smooth surface? We could obviously approximate any surface defined in terms of NURBS or Subdivision by simply including lots of triangles/quads in a normal mesh. Is the reason we don't to avoid representing a huge number of triangles?
Are there advantages to simpler interpolation techniques, particularly when it comes to running time? I would assume that it may not always be desirable to work with a complicated basis, when hardware is very well-suited to doing work with lines.