Definition of Ito integral: adapted vs predictable processes

Question

In Öksendal he defines the Ito integral for adapted processes, and everything works out smoothly. However, my professor tells me that in much of the standard litterature, e.g. Williams: Diffusions, Markov Processes and Martingales vol 2, the integral is only defined for predictable processes, a class strictly smaller than that of adapted processes. My question is then, why? What is the practical difference?