This paper is concerned with the differential sensitivity analysis and the optimal control of evolution variational inequalities (EVIs) of obstacle type. We demonstrate by means of a counterexample that the solution map S of an EVI with a unilateral constraint is typically not (weakly) directionally differentiable or Lipschitz continuous in any of the spaces H-s(0, T; H), s >= 1/2, where (0, T) is the time interval and H is the pivot space of the underlying Gelfand triple V hooked right arrow H hooked right arrow V*. We further establish that, despite this negative result, the solution operator is always strongly Hadamard directionally differentiable as a function S : L-2 (0, T; H) -> L-q (0, T; H) for all 1 <= q <= infinity, weakly-* directionally differentiable as a function S : L-2 (0 , T; H) -> L-infinity (0, T; H), and weakly directionally differentiable as a function S : L-2 (0, T; H) -> L-2 (0, T; V). Using the differentiability properties of the map S, we derive strong stationarity conditions for optimal control problems that are governed by EVIs of obstacle type. The resulting optimality system is compared with that obtained by regularization.