On the basis of a minimax-maximin differential game of minimal deviation of a moving object from a given target on a given time interval, we discuss the relationship between the unified form of the game and the interpretation of this game in terms of Subbotin's generalized minimax solution of Hamilton-Jacobi equations. In addition, we consider the relationship between the chosen formalization of a differential game and investigations of this game on the basis of solutions of parabolic equations degenerating into Hamilton-Jacobi equations as the diffusion term tends to zero. The related generation of minimax and maximin controls with a stochastic guide is described. We analyze the similarity between the unified form of a differential game and the concept of differential game suggested by Pontryagin.