The information matrix equality makes the Fisher doubly useful: it can be computed from first derivatives alone (an expectation of squared gradients, so it is always nonnegative), yet it estimates the Hessian of the negative log-likelihood at the optimum. In classical statistics its inverse lower-bounds estimator variance (the Cramér-Rao bound) and yields standard errors for maximum likelihood estimation.
In deep learning it appears wherever curvature is needed cheaply: natural gradient methods precondition updates with it, and elastic weight consolidation uses its diagonal as a per-weight importance score, penalizing movement of exactly the weights whose disturbance would most change the old task’s predictions.