Clamped and Natural Cubic Splines
The “natural” cubic spline is defined to be the one that has zero second derivatives at its end points. Your given polynomial $p$ probably does not have zero second derivatives at its end points, so it can’t coincide with the natural cubic spline.
I say “probably” because if $p$ is a constant function, then it will have zero second derivatives at its end points, and then it will coincide with the natural cubic spline.
To show that $p$ coincides with its clamped spline approximation, think about the fact that spline interpolation problems have unique solutions.