This paper deals with optimal control problems for dynamical systems governed by constrained functional-differential inclusions of neutral type. Such control systems contain time delays not only in state variables but also in velocity variables, which make them essentially more complicated than delay-differential (or differential-difference) inclusions. Our main goal is to derive necessary optimality conditions for general optimal control problems governed by neutral functional-differential inclusions with endpoint constraints. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of endpoint constraints. Developing the method of discrete approximations (which is certainly of independent interest) and employing advanced tools of generalized differentiation, we conduct a variational analysis of neutral functional-differential inclusions and obtain new necessary optimality conditions of both Euler-Lagrange and Hamiltonian types.