I am dealing with a numerical problem with cubic spline, but I am a little bit confused while using them because of terms spline and b-spline.
In simple words, what is the difference between the cubic spline and cubic b-spline? Are these both terms the same, or is the b-spline another name for cubic spline?
A spline is a curve that is formed by stringing together polynomial pieces in a clever way (so that continuity between the pieces can be controlled). The polynomial pieces can have any degree. A common choice is degree = 3, in which case the spline is called a "cubic" spline.
Any spline (of any degree) can be represented in b-spline form. From a mathematical point of view, this is because b-splines can be used to construct a basis for any spline space.
So, to answer your question: a cubic b-spline is one possible way to represent a cubic spline, just as "XVI" is one way to represent the number sixteen.