Some theory. There are theorems which show that the accuracy
of spline interpolation is better than polynomial interpolation
in the sense that the interpolant more closely matches the
(unknown) function from which the data is derived, across the
interval where the data lies. For example, see Equation (2) on
page 164 of our text (error estimate for polynomial
interpolation) and the theorem on linear splines on page 322 of
the text.
The bottom line is that, for sufficiently smooth functions
f(x), the maximum difference between f(x) and the piecewise
linear (spline) goes to zero like O(h2), where h is
the interval size. We can make no similar statement about the
error shrinking as n grows for polynomial interpolation (it may
not even shrink at all, in some cases). For cubic splines, the corresponding
theorem says the error shrinks like O(h4).