Here are some applets that demonstrate different kinds of splines.
Bezier splines pass through only the end control points. Bezier splines suffer from the following drawback. If you are defining a curve with more control points you have two options. Increase the degree of the curve, or find a way to stitch the curves together. Increasing the degree of the curve is not desirable for two reasons. One, it is harder to evaluate the curve because there are more terms to consider. Two, it lacks local control, that is moving any control point affects the whole curve.
Stitching Bezier curves together by hand is a pain, and ideally, it is something that you'd like the computer to just do for you, this is where the other spline definitions come into play.
B-Splines rarely interpolate any of the control points. But, they stich together with the highest amount of continiuity (C2) that a cubic polynomial can have.
For those that find non-interpolating splines non-intuitive, Catmull-Rom splines are nice. They interpolate all the points except the end-points. They stitch together automatically, but they do not offer as much continuity (C1) as B-Spline curves.