The Akima interpolation is a local fit, requiring information only about points in the vicinity of the interval being interpolated to define the coefficients of the cubic polynomial. This means that each data point in an Akima spline only affects the nearby portion of the curve. Because it uses local methods, an Akima interpolation is calculated very quickly.
Akima produces good results for the value of the function being approximated. AKISPL returns good estimates for the first derivative of the approximated function when the data points are evenly spaced. In instances where the data points are unevenly spaced, the estimate of the first derivative may be in error. In all cases, the second derivative of the function being approximated is unreliable.
The cubic interpolation is a global fit. Global methods use all the given points to calculate all the coefficients for all the intervals in question simultaneously. Therefore, each data point affects the entire cubic spline: if you move one point the whole curve changes accordingly, making a cubic spline rougher and harder to force into a desired shape. This is especially noticeable on functions with linear portions or sharp changes in the curve. In these cases, a cubic spline is almost always rougher than an Akima spline.
Both global and local methods work well on smoothly-curving functions.
CUBSPL, though not as fast as AKISPL, produces good results for the value of the function being approximated, as well as its first and second derivatives. The data points don't have to be evenly spaced. The solution process often requires estimates of derivatives of the functions being defined. The smoother a derivative is, the easier it is for the solution process to converge.
Smooth (continuous) second derivatives are important if you use the spline in a motion. The second derivative is the acceleration enforced by the motion, which defines the reaction force required to drive the motion. A discontinuity in the second derivative means a discontinuity in the acceleration and therefore in the reaction force. This can cause poor ADAMS/Solver performance or even failure to converge at the point of discontinuity.
The B-spline interpolation method is primarily designed to describe 3D geometric curves. Although the B-spline can be useful for geometric applications, you should use AKISPL or CUBSPL to construct most motions, forces, and other such entities.