For some applications, getting the solution is most important. For other applications, the sensitivity of the solution to its parameters is more important. This is also a controversial issue. Certainly, a global solution is always desirable, but one has to pay additional cost for it. In some engineering applications, such as the inverse problem in the imaging of brain function, a global solution is required because the underlying activity is widely distributed.
Many applications require a global solution.
For example, to apply lower bounding techniques for combinatorial problems, global solutions are needed, i.e., relaxations of hard combinatorial problems result in continuous optimization problems where only a global solution provides a bound. Theoretically, global optimization of many nonlinear problems cannot be done in polynomial time. The problems have to be small. Besides, there does not seem to exist a practical stopping rule for global optimality. An NLP text in the style of Bradley, Hax and Magnanti is a start.