Moving a control point of a b-spline curve will affect a few nearby segments, not just one, but will generally not impact the entire curve.
Typically, people who design high-quality surfaces (like the exterior surfaces of car bodies) prefer Bézier curves and surfaces. They are suspicious of the joints where the segments of a b-spline curve connect together. If the b-spline surfaces are cubics (as they often are), then the suspicion is justified, because the joins will be visible in surface reflections. For these sorts of folks, the local nature of b-spline editing is a disadvantage, because it destroys the "character" of the curve and often leads to local undulations and even inflexions.
On the other hand, if you're doing approximation, then b-splines are probably a better choice, in many situations. The only way to get more degrees of freedom (and a better approximation) with Bézier curves is to increase the degree. A string of low-degree segments (i.e. a spline curve) is often preferable to a single segment with high degree. Mostly because this makes evaluation of curve points and derivatives less expensive.
People will tell you that high-degree polynomials (i.e. Bézier curves) are a bad idea because they tend to "wiggle". This is numerical analysis folklore, but according to Trefethen and Higham (who ought to know), the bad rep is undeserved, and the problems are avoidable. Anyway, there is no chance that high-degree curves defined by Bézier control points will oscillate badly -- the variation-diminishing property of Bézier curves guarantees this. So, no need to be afraid of the wiggling horror stories.
And, in case you're wondering, NURBS (rational curves, as opposed to polynomial ones) are now considered to be a bad idea even by the people who first proposed them.