While they are frequently found embodied in the same character, it need not be the case that a villain and an Antagonist are not occupying the same space in a story. A villain is a character who represents a choice of moral evil in a story, while an antagonist is an element of the story that directly challenges the hero and his/her goals and stands as an obstacle to the hero's success.
As I have said in other answers, the Disney Film Mulan (1998) has the villain of Shan Yu, one of the weakest villains in the animated canon of Disney films, but he does not directly oppose Mulan, nor is his defeat a goal of hers. Rather, Mulan's stated goal is finding a place in her society, when her personality does not fit China's gender roles. Thus Shan-Yu's invasion is neither done by his motivation to defeat Mulan nor is her decision to hide that she is a woman to join the army motivated by a personal animosity to Shan Yu. I don't believe she even mentions his name until the Third Act of the film.
Rather, Mulan's antagonist is Chinese society as a whole, which freely admits she is a second class citizen because of her gender, and her victory over the antagonist is when she saves the Emperor only after Shan Yu kidnaps him. Shan Yu for his part, doesn't begrudge Mulan for her gender, but drops more valuable targets because Mulan is "The soldier from the mountains" who single handedly looked him in the eye and defeated his 1000s strong army. That is, he saw her as a threat to his present safety, and knew she was a dangerous tactician. He never once refers to her by her gender, something the very allies she saved held against her despite her quick thinking saving their asses. In fact, by the time she and Shan Yu face off, the Emperor's already safe.
In another example, in some works where the villain is the protagonist (Breaking Bad) the antagonist is the hero or a character who represents a choice of moral good in the story. In Breaking Bad, Walter White is the villain protagonist and and is ultimately confronted by Hank Schrader, who despite his personal flaws, is committed to stopping Walter's drug empire, making him a heroic antagonist.
In some stories, the line is further blurred. In A Christmas Carol where the character's are all concerned with Scrooge's salvation, Scrooge is our protagonist, and while he is clearly evil on the onset, the spirits mission is to show Scrooge that much of his pain is due to no one but himself: If he insists on blaming others in the past for his self-induced misery, and refuses to see goodness when others offer it to him despite his abrasiveness, he will die unloved and un-mourned. In this case, Scrooge is the protagonist, and his miserly and cruel personality in the beginning is his own antagonist the Ghosts are offering him a chance to see the error of his ways, but it's his own ego that he has to beat.
Still in other works, the "antagonist" is not evil at all and is merely doing what it does because it is in it's nature to do it. In the film, Homeward Bound, while the Dogs and Cat are clearly the protagonists of the film, their is no one who means them ill will and all "antagonists" are trying to help them if they are human and the animals do not understand this OR are environmental and doing what they would do. The Mountain Lion is hunting the dogs, but it's only doing so because, as a wild animal, it hunts for it's food. The river sweeps the cat away and injures her out of no malice. It behaves as a river would. In these cases, no villain exists, but the antagonist exists in the form of the neutrality of the environment. In these cases, the protaganists simple goal of "continue living" is opposed by nature doing what it does, and the hero must overcome by using his wits and skills and quick thinking to last while out of his element.
An antagonist doesn't need to have any personal agency in the story or have a strong reason to oppose the protagonist or even hate the antagonist. They don't have to be characters or personified. Only villain antagonist does need some reason to oppose the hero because it is their morally evil actions that provide a challenge for the hero to overcome.