The paper continues the studies started by the authors in the equivalence of the measures generated by the solutions of nonlinear evolution differential equations with unbounded linear operators perturbed by random Gaussian processes in a Hilbert space, in particular H. Two different nonlinear evolution differential equations perturbed by the same random Gaussian process in the right-hand side are considered in the space H. The sufficient existence and uniqueness conditions are established for the solutions of these equations, the equivalence of the measures generated by the solutions is proved, and explicit formulas of the Radon - Nikodym density of the respective measures calculated in terms of the coefficients of the considered equations are written.Nonlinear evolutionary differential equations with unbounded linear operators, disturbed by Gaussian random processes, are considered in an abstract Hilbert space. For the Cauchy problem of the differential equations under study, the sufficient existence and uniqueness conditions for their solutions and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon - Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.