Perhaps a personal rivalry could be the cause. They are both strong and skilled warriors. There could be an underlying competition between them for the title of the greatest warrior in the area. This could manifest as a fight, especially if there was some sort of public display or competition where their egos were at stake.
It could also be due to differences in their beliefs. Brasidas is a Spartan, with his own set of values and loyalty to his city - state. Kassandra, on the other hand, has a more individualistic view having traveled far and wide. Their different outlooks could lead to a clash. For example, if there was a decision to be made about how to deal with a new threat in the region, Brasidas might want to handle it in a traditional Spartan way, while Kassandra could have a more unorthodox approach.
It's really hard to say. Brasidas is a formidable Spartan warrior with great strength and military training. However, Kassandra has the advantage of being a misthios, with a wide range of skills and experiences from all over the world. If the fight was in a more traditional, open battlefield, Brasidas might have the upper hand due to his experience in phalanx warfare. But if it was in a more urban or diverse environment, Kassandra's adaptability could give her the win.
It could be due to a misunderstanding about their respective roles in their group or in a mission. For example, if Sirius thought James was taking too much of a leadership role without consulting him, it might cause a rift. In fanfiction, the writer might explore how their strong personalities clash in such situations.
Another reason might be the stress of their quests. They are constantly under pressure to save the world or their friends. This stress can build up and cause them to lash out at each other. For instance, if a quest isn't going well and they keep hitting dead ends, they might start blaming each other for mistakes made along the way.
In some fanfictions, their fight might be due to a competition at Hogwarts. Maybe they are both vying for the top spot in a particular class or for a special award. Hermione's studious and competitive nature combined with Draco's pride and desire to outdo others could easily result in a fight.
One reason could be jealousy. Maybe Nico is jealous of Percy's popularity at Camp Half - Blood. Another reason might be a difference in their goals. Percy wants to protect the camp and the mortal world in a certain way, and Nico has his own ideas that don't align with Percy's. So, this divergence of interests could lead to a fight.
One reason could be that they might have different interpretations of Harry's character in the fanfiction. Hermione, being logical, may see Harry in a more practical light, while Ron, with his emotional side, could have a different view.
One possible reason could be their rivalry over a love interest, like Bella. Jacob loves Bella and Edward is in a relationship with her. This love triangle often leads to tension and could easily erupt into a fight.
One reason could be a misunderstanding about their family's future. Maybe Edward has a vision of how the family should evolve that Carlisle disagrees with. For example, Edward might want to expand the family in a different way than Carlisle.
Misunderstandings about their future plans often lead to fights. Percy might want to focus on one type of adventure while Annabeth has other ideas. They argue about it, but in the end, the love between them overcomes the differences and they kiss. This shows how their relationship is complex and full of different opinions that need to be resolved.
It could be a rival assassin. They might see Kassandra as a threat to their own reputation or contracts.