When is cubic spline interpolation better than an interpolating polynomial?
The phenomenon being discussed is Runge's phenomenon.

The maximum absolute value of the $n$th derivative of $\sin(\omega t)$ is $\omega^n$. For Runge's function $\frac{1}{25t^2+1}$ the maximum absolute value of the $n$th (even) derivative is $5^nn!,$ where $n!$ denotes factorial. This is much faster growth. Only if the derivatives grow too fast by increasing $n$, then it is possible that interpolation error diverges as the interpolation order is increased. Exponential in $n$ is not yet too fast. Have a look at: James F. Epperson, On the Runge example, The American Mathematical Monthly, vol. 94, 1987, pp. 329-341.

If a function has only continuous derivatives, then the competing approach, piece-wise polynomial spline interpolation always converges if a small fixed number of its early derivatives are bounded over the interval of interest, see Wikipedia article on linear interpolation as an example.

If both methods converge, then (non-piecewise) polynomial interpolation has the advantage of a higher polynomial degree if many samples are used, and may provide a better approximation, as you saw in your sine example. You may also be interested in L.N Trefethen, Two results on polynomial interpolation in equally spaced points, Journal of Approximation Theory
Volume 65, Issue 3, June 1991, Pages 247-260. Quote:


  [...] in band-limited interpolation of complex exponential functions 
  $e^{i\alpha x}\, (\alpha \in \mathbb{R}),$ the error decreases to $0$ as $n \to \infty$ if and only if $\alpha$ is
  small enough to provide at least six points per wavelength.


You have 6.5 samples per wavelength.