The two most beloved figures of DC Comics' massive stable of superheroes are also two of the most different characters in their beloved canon. One of the least creative and most pervasive ideas in the history of their depiction centers around pitting the two heroes against each other, and it reveals a lot about both.
Batman and Superman are two of the three key members of the Justice League, the third being Wonder Woman. Clark Kent was essentially born a god and strives to bring his shining optimism to the world with unending heroism. Bruce Wayne is a mortal man motivated by his life's lowest moment to pursue justice from the shadows. They're different, but they want the same things. It's a classic trope.
In some comics, Bruce and Clark disagree on exactly how to pursue the generally good ideals both men want. In others, that philosophical schism evolves to the level of outright conflict. In still others, they simply don't care for each other, and swiftly descend to the level of begrudging teammates if not outright enemies. All of these examples, however, are in the minority. In the overwhelming majority of comic books which feature both Superman and Batman, each holds the other in the highest possible regard. Superman has undying respect for a normal man giving everything to protect his city, world, and species. Batman considers Superman the paragon of virtue, an alien capable of anything who chooses to devote himself to good. Both regularly consider the other the better man. They have very little in common, yet each sees the other as the ultimate hero. What they share is more than friendship, it's an unbreakable bond. Unfortunately, those who have only seen the recent films would not agree with this.
The first and only filmmaker to put Batman and Superman on the big screen together is Zack Snyder. Their first outing is Batman v. Superman: Dawn of Justice, a film literally named after their conflict. Snyder's Batman is The Punisher in a cowl, and his Superman is a religious metaphor who grows furious every time he's questioned. Batman hates Superman because of the collateral damage caused in the final battle of Man of Steel. Superman hates Batman because he's been viciously murdering criminals.
Neither character is particularly comparable to their comic book counterpart, making their battle against each other feel like two new characters needlessly squabbling. A bigger threat is running around in the background, and they come together, in the end, to defeat it, as everyone guessed they would. By the next film, Justice League, Batman decides he was wrong about his previous nemesis and endeavors to bring him back from the dead. It's a terrible showcase for Clark and Bruce, and an even worse one for their relationship.
There are a few DC Animated films that put Clark and Bruce together. There are two based on Jeph Loeb's Superman/Batman comic book series, which takes a more nuanced approach to the duo's relationship. Several Justice League films see the two interact, but typically separate them or put very little focus on them as a duo. More common are pieces like Injustice, which depicts an evil Superman, or Doom, which concerns a villain with a plot that's all Batman's fault. These films are less well-known than the live-action efforts, but they also rarely depict Clark and Bruce as allies.
Pre-Crisis DC Comics depicted Bruce and Clark as uncomplicated best friends. The turning point came in Frank Miller's 1986 series The Dark Knight Returns, which depicted Superman as a government pawn and an older Batman as the last hero who could stop him. Like Alan Moore's Watchmen, Dark Knight Returns is one of the most influential comic books ever written, with mostly negative impact. It's mostly down to Miller that the default version of Batman is the comically grim loner, and the default version of Superman is either a joke or the bad guy.
The hack take on Batman is to eliminate all other heroic influences, regardless of how important they are. No sidekicks, no allies, no teammates. Anything as uplifting as friendship or camaraderie would distract from his brooding and misery. The hack take on Superman is that his uncomplicated boy scout morality and godlike power make him boring. Playing him straight would alienate people who like their superhero stories to be "mature," so filmmakers choose to either let him kill countless people or subvert him by making him evil. These are both overdone tropes of the character that have largely ruined modern depictions. It all stems back to the self-defeating urge to make a story about costumed crimefighters into something that it isn't.
The bitter hatred between Superman and Batman is often pointless, but it's a symbol of the larger problems with their modern portrayals. It was once transgressive to depict the two heroes as enemies. It would be groundbreaking today for a filmmaker to let them be friends.