First, here's the video. Characters Pearl and Amethyst have a . . . well, a special kind of relationship. (This AMV is set to the song "What Is This Feeling?" from the musical Wicked.)
The song fits them well because they're very much like Elphaba and Glinda, the characters in the musical: they fundamentally hate certain things about each other and are extremely prone to squabbling at the drop of a hat.
And yet, after some realizations, some intense conflicts, and some bonding experiences, they become very good friends and depend on one another.
In romance, we have this trope where characters hate each other and then realize they are madly in love with each other and proceed to have a passionate romance full of fireworks. But it's just as common with friendships in stories, and we like seeing these kinds of opposites-attract characters throw punches. Especially when it leads to them finding common ground.
So what exactly is this relationship? Do characters have to change as people before they can find this equilibrium? Well no. Of course not. And they're not even necessarily compromising when they begin to see eye to eye. I think, for this to work, characters who squabble and later become close often dislike something about the other character that has its roots in something they don't like about themselves.
For instance, take Galinda and Elphaba. (For those who haven't seen the musical Wicked, Galinda/Glinda is the eventual "Glinda the Good" and Elphaba is the "Wicked Witch of the West," derived from Oz stories, but this is a prequel of sorts, beginning when the characters are in college.) Galinda starts school as an aspiring magic student from an upperclass family, and she's forced through weird circumstances to share a room with Elphaba, who has been chosen over Galinda for a magic apprenticeship. Elphaba also refuses to be the meek "different" girl even though she's bright green and intellectual, so she's pretty aggressive and tries to be proud.
They pretty much instantly loathe one another.
Galinda is popular, pretty, rich, and not too happy about not getting what she expected. And she feels like she works hard and makes sacrifices to remain well-liked and successful, so someone like Elphaba winning the apprenticeship irritates her. And Elphaba may be a misfit in many ways, but she knows what she's good at, and she has spent much of her life trying not to be ashamed of who she is--so obviously a popular girl deliberately leading a charge against her is worthy of intense rivalry.
But after they've bonded somewhat, they do realize they want similar things and care about similar causes, though they continue to have different ways of approaching those things. Glinda admires Elphaba's brash refusal to compromise and willingness to lay her comfort and her life on the line for what she believes, and Elphaba admires Glinda's ability to work within the system and use her charisma to effect change. Neither wants to be what the other is, but they do accept the inherent value in each other's belief systems, and they care about each other as people once they've each stopped seeing the other as a symbol of something they themselves are lacking.
As for the Steven Universe characters Pearl and Amethyst, we don't yet know how they met or what their dynamic was like at that time, though a couple flashbacks from their younger years have shown absolutely no squabbling--if anything Pearl just kind of treated Amethyst like a child and bossed her around a little bit, but she didn't seem to mind.
But based on how we originally meet these characters, they have an antagonistic relationship that reads a bit like sibling rivalry--Pearl as the perfect older one, Amethyst as the screwed up younger one. (Pearl is a LOT older than Amethyst--we don't know how much older, but context suggests she is thousands of years older. She once says she was "only" a few thousand years old when she started fighting in her army, and Amethyst wasn't born yet when their war started.) On the surface, they hate each other because Pearl likes neatness and order and responsible behavior while Amethyst likes messes and chaos and lazy behavior. But that's not the half of it with these guys.
Pearl is not perfect at all, but she clings to that perfect image because keeping things clean and orderly is one of the only things in her life that she can control. She is one of the most broken characters I've ever seen--she's way more of a mess than Amethyst, despite images--and I think that's one of the things she's envious of Amethyst for. Amethyst isn't an anxious person--maybe a little restless sometimes, but not anxious--and her life and history is a lot simpler than Pearl's. She knows how to have fun. I'm not sure if I've ever actually seen Pearl legitimately enjoy herself (with a couple exceptions that also led to pretty awful things that I won't talk about here). She's always just trying to stay afloat and manage her nerves, and Amethyst is a good scapegoat. If you can blame someone for driving you up the wall, you don't have to take ownership of your grief, traumatic stress disorders, and anxiety.
Amethyst kind of buys the "Pearl is perfect" ruse (at least until she sees enough of her breakdowns and processes what they mean). She feels like the oddball in their group--she missed the war everybody else fought in when they saved the planet thousands of years ago; she was born on Earth while her teammates were born on another planet; she's very physically strong but the others are better fighters than she is; she's the only one who enjoys human behavior like sleeping and eating; and she was essentially created by the bad guys in the story and she carries guilt over being an instrument of destruction. Plus if they were on their planet of origin, she would outrank Pearl and everyone on their team because amethysts are quartzes and quartzes are apparently among the biggest badasses in the galaxy, but here on Earth she gets bossed around by everyone. And then there's Pearl telling her she's a mess, she's irresponsible, she's ruining everything. So she takes some comfort in rebelling--sometimes because she needs an outlet for her aggression, and other times just to do it.
But even though they have different interests and values, these two are on the same team. (As their leader, Garnet, sometimes has to remind them explicitly.) They both care about protecting the planet and preserving the lives of those in their care--and they've both been hurt by some of the awful traumatic things that have happened to them (most notably the death of one of their teammates; assuming I'm reading the context right, Amethyst lost a mother figure and Pearl lost a romantic partner when Rose Quartz died, and they both lost stability and direction in this world). Sometimes they even fight over who owns that grief or who has more of it, but they can agree that they miss her terribly. They both want to be important to the team. They both want respect from Garnet. And they've both realized that in a world where neither of them has fulfilled the purpose for which they were created, they are in a similar struggle to determine what they value and who they are. People with similar struggles can see each other as competitors, but they can also find comfort in each other.
Audiences love following characters who have authentic conflicts, but we will start to find it tiresome if it never goes anywhere. It can build to some resolution--explosive or quiet, doesn't matter--and while it can be satisfying to watch diametrically opposed characters like hero vs. villain fight to the death, it can also be fulfilling to see them become better companions. Building fundamental differences into characters that are easy to see on the surface is an ingenious way to hint at the real issues they have with themselves, and it's incredible when they can learn to help each other through a conflict when they might have been on opposite sides earlier in their evolution.
Conflict in the past makes it that much more beautiful when the characters have moved through it into what they can truly share.