I have read in a number of places that cubic splines are of more practical use than quadratic splines in general (there are exceptions of course). Anyone know specifically why they are more applicable/better?
A cubic curve can twist in space (i.e. it can be non-planar). A quadratic curve is just a parabola, so it's always planar.
In real applications like graphic arts, engineering, manufacturing, nobody cares much about derivatives, they only care about curvature. And it's possible to get continuity of curvature without continuity of second derivatives (so-called G2 splines, versus C2 ones). So the C2 argument for cubics is a bit fragile.
For some applications, like design of car bodies or cams, cubic splines are not good enough, because you need continuity of the derivative of curvature (G3 continuity). So, in these applications, higher degree curves are often used.
In many situations, splines are used as approximations of more complex functions. If you use cubics, then you'll need fewer polynomial segments for a given approximation tolerance. But each segment will be more complex, so more difficult to handle in subsequent calculations. That's basically the trade-off -- number of segments versus complexity of segments.