The improvement is that the function can represent any plot by adjusting a single coefficient, whereas a polynomial approximation has a number of coefficients equal to the number of points in the plot.

This is only an improvement in a superficial, linguistic sense though. If the single coefficient is just a bit-packing representation of many more degrees of freedom (because of its huge precision), then from a model information complexity point of view, the polynomial model could actually have fewer parameters, in the sense that the overall size of the combined parameter space is smaller, e.g. it’s a smaller program size.

It reminds me of the Grue vs. Bleen question in philosophy.