The truth is that antagonists are not always villains, villains are rarely the only antagonists, and while not every story has a villain, almost every story has at least one antagonist.
Right now you might be thinking, "Whaaa-???" and scratching your head.
Allow me to explain.
Many sources, including the Merriam-Webster dictionary, simplify the relationship between hero and villain, protagonist and antagonist. The villain/antagonist is the one who opposes the hero/protagonist.
This is most clearly seen in archetypal children's stories of Good vs Evil. Good Red Riding Hood vs Big Bad Wolf. Innocent Hansel and Gretel vs Bad Cannibalistic Witch. The opposition of good hero against evil villain is black and white.
But very little, if anything, is really that simple.
What about an opponent who isn't evil? Here I'm thinking of a story such as The Fugitive (the 1993 movie version with Harrison Ford and Tommy Lee Jones).
In The Fugitive, [*Spoiler alert!] Dr. Richard Kimble is falsely convicted of killing his wife. When Kimble escapes on his way to prison, U.S. Marshal Samuel Gerard is brought in to capture him.
As far as Gerard is concerned, Kimble is guilty and is literally an escaped convict. Gerard is on the side of justice and is doing his job. In short, he's not a villain.
So what is his role in the story?
Every protagonist, whether hero, antihero, or something else, has a goal. There is something the protagonist wants to obtain or achieve. This goal, and the protagonist's desire for it, drives the story. It might be a geographical destination, a job, an object, a romantic partner, a role in a play, a spot on a team, revenge for a great wrong or a slight, protecting a loved one or the entire known world.
Whatever it is, there's a path the protagonist must take to achieve a goal.
The antagonist stands squarely in the protagonist's path toward that goal.
In The Fugitive, Kimble uses his escape to find out who murdered his wife and lead Gerard to that person so justice can be served and his own name be cleared. Because Gerard is hunting him, Kimble cannot take a direct path toward his goal, but is forced to get creative and pursue his goal in roundabout ways.
Though they seem to be working at cross purposes, both Kimble and Gerard ultimately want the same thing, namely whoever murdered Kimble's wife to be captured and sent to prison.
As it happens, there is a villain in The Fugitive. The villain is whomever actually killed Kimble's wife, but the killer's identity isn't what drives the story. In fact, it doesn't really matter who the killer is, as long as it makes sense in the end. What matters, what's central to the story, is that the killer isn't Kimble.
But, if Gerard wasn't chasing him, and Kimble was simply free to find the killer at his leisure, there would be no challenge, no suspense, and no emotional investment on the part of the audience.
An antagonist blocks the protagonist's way forward, diverting him from his path. Because of this, the protagonist is often forced to grow, discover new internal resources, become braver, cleverer, or somehow better than before.
A villain can do that, too, but as we've seen, not every villain is the main antagonist and not every antagonist is a villain. An antagonist can be a devoted lover, a worried parent, a well-intentioned stranger, or a simple-minded innocent unaware of the consequences of her actions, who throws a monkey wrench into the works.
In epic tales like Harry Potter, Lord of the Rings, Star Wars, and Game of Thrones, there are multiple antagonists. The longer and more arduous the protagonist's journey, the more obstacles they're going to encounter.
So the next time you're reading, watching, or writing a story, ask yourself, "Who is the villain? Who is the antagonist?" They may not be the same person.