For the purposes of this answer, let us say that the two sides are President Franklin D. Roosevelt on the one side and the anti-New Deal forces on the other. Both sides did, indeed, both win and lose.
Roosevelt lost in two ways. First, he did not get to “pack” the Supreme Court as he had wanted to. In addition, his own prestige took a major hit. He was seen as having overreached and as having to tried to take more power than the Constitution grants the president. It was in these ways that his opponents won. They prevented the change to the Court and they weakened Roosevelt.
But President Roosevelt won in a very important way. The whole reason that he wanted to pack the Court was because its justices had been striking down his New Deal legislation. The court packing plan helped to reverse this trend. It is said that the Court executed the “switch in time that saved nine.” It reversed itself and started to allow New Deal programs to stand. In this way, Roosevelt won and his opponents lost; the New Deal would be allowed to proceed without judicial interference.
One of the other reasons that the Court reversed itself in the aftermath of the court-packing fight was that most of the more conservative justices (and indeed some that were not so conservative) retired in relatively short order. Starting with Willis Van Devanter, who retired in 1937, eight of the nine Supreme Court justices retired before FDR's death, and were replaced with his nominees. This, in addition to the fact that Owen Roberts, and to a lesser extent Charles Evans Hughes, switched his position on the New Deal programs, meant that the Supreme Court represented less of a threat to his programs than they had been.