Let's say your hero is a comfort-loving hobbit (The Hobbit) or a safety-loving fish (Finding Nemo). Their antithesis world will be a world full of danger and privations. For Bilbo, it's the world outside the Shire; for Marlin, it's the open ocean.
But let's say your hero isn't going on a literal journey. He's going to spend most of the story in one place. In that case, his old world will become his new world as it crumbles around his ears. In Iron Man 2 Tony Stark's arc reactor is slowly poisoning him, he's alienated his best friend and his love interest, and an old nemesis from his past shows up to kill him. He has no control over this new world and his efforts at fixing his problems only make them worse.
The new world is a world full of new experiences and challenges for the hero. For the first few chapters he's probably going to spend most of his time adapting to his new surroundings, trying to figure out the rules of the new “world”, and generally attempting to survive in an unfamiliar setting. From the catalyst moment on, the hero is on a journey back to normalcy--to what his world was back at the beginning, although neither your hero nor his world will ever be entirely the same again.
By the midpoint of the story your hero will more or less have things figured out. He'll have "mastered" (to a degree) his new world and learned how to use it to his own advantage. As the story heads towards the climax, it moves into what Blake Snyder calls the "synthesis world" where the hero uses what he's learned from both his new and old worlds as he takes on the bad guys and problems of the story.
At the end of the story, the hero is in a final new world--this world is often similar to the one he started in, but of course now the problems are solved, the bad guys are defeated, and the hero has all the lessons he learned in his antithesis world to pilot him through his future.