Why was John Nash's 1950 Game Theory paper such a big deal?
I think von Neumann dealt with the case $n=2$, and it was by no means obvious how to extend the concept of equilibrium for the general case and prove that it always exists. More precisely, $n$ players before Nash were reduced to the $n=2$ case by partioning the players into two groups in all possible ways. Once you regard several players as a single player, they are meant to cooperate as they must act like a single player. Nash is very clear about this in his 1951 Annals paper:
Von Neumann and Morgenstern have developed a very fruitful theory of two-person zero-sum games in their book Theory of Games and Economic Behavior. This book also contains a theory of $n$-person games of a type which we would call cooperative. This theory is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game.
Our theory, in contradistinction, is based on the absence of coalitions in that it is assumed that each participant acts independently, without collaboration or communication with any of the others.
The notion of an equilibrium point is the basic ingredient in our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing "good strategies." In the immediately following sections we shall define equilibrium points and prove that a finite non-cooperative game always has at least one equilibrium point. We shall also introduce the notions of solvability and strong solvability of a non-cooperative game and prove a theorem on the geometrical structure of the set of equilibrium points of a solvable game.
This answer overlaps with other answers but I think another restatement may be helpful because the situation is slightly confusing.
After the two-person zero-sum result, it is natural to ask about extending the results to $n>2$ and to non-zero-sum games. Sometimes it is stated that Nash was the first to carry out this extension, but this is slightly misleading, because von Neumann and Morgenstern did consider both $n>2$ and non-zero-sum games and proved various things about them. However, the key point is that it's important to ask the right question. Intuitively, the basic question in game theory is to find the "optimal strategy", but it's not immediately clear what this means in an $n$-person non-cooperative game. We now understand, thanks to Nash, that a basic necessary condition for a set of strategies to be "optimal" is for them to form a Nash equilibrium, but von Neumann and Morgenstern did not hit on this concept. When they treated $n$-person games, they addressed different questions, such as what happens if the players form two coalitions. So Nash didn't just answer the obvious question; the right question wasn't obvious, but he found it anyway, and answered it.
The second innovative aspect of Nash's work is that the two-person zero-sum result was based on the theory of linear programming and minimax. Proving the existence of a Nash equilibrium requires different techniques. So the naive approach to generalization, namely staring at the existing result and trying to figure out how to use the same ideas to prove something more general, does not lead to Nash's key insight.
The significance is best interpreted in conjunction with Nash's accompanying work.
Myerson gives a good history of the theory: Nash equilibrium and the history of economic theory
Here are some important points:
Thus von Neumann (1928) argued that virtually any competitive game can be modeled by a mathematical game with the following simple structure: There is a set of players, each player has a set of strategies, each player has a payoff function from the Cartesian product of these strategy sets into the real numbers, and each player must choose his strategy independently of the other players. ...
Von Neumann did not consistently apply this principle of strategic independence, however. In his analysis of games with more than two players, von Neumann (1928) assumed that players would not simply choose their strategies independently, but would coordinate their strategies in coalitions. Furthermore, by his emphasis on max-min values, von Neumann was implicitly assuming that any strategy choice for a player or coalition should be evaluated against the other players' rational response, as if the others could plan their response after observing this strategy choice. Before Nash, however, no one seems to have noticed that these assumptions were inconsistent with von Neumann's own argument for strategic independence of the players in the normal form.
Von Neumann (1928) also added two restrictions to his normal form that severely limited its claim to be a general model of social interaction for all the social sciences: He assumed that payoff is transferable, and that all games are zero-sum.
In contrast, Nash provided a way to deal with the more general problem of non-transferable utility and non-zero-sum games.
But the most important new contribution of Nash (1951), fully as important as the general definition and the existence proof of Nash (1950b), was his argument that this noncooperative equilibrium concept, together with von Neumann's normal form, gives us a complete general methodology for analyzing all games.... Von Neumann's normal form is our general model for all games, and Nash's equilibrium is our general solution concept. ...
Nash (1951) also noted that the assumption of transferable utility can be dropped without loss of generality, because possibilities for transfer can be put into the moves of the game itself, and he dropped the zero-sum restriction that von Neumann had imposed.
- Reference: Myerson, Roger B. "Nash equilibrium and the history of economic theory." Journal of Economic Literature 37, no. 3 (1999): 1067-1082.