It could also be that Snape represents everything that Harry hates about Hogwarts at that moment in the fanfiction. Snape's strictness, his partiality towards certain students, and his general unfriendly attitude might make Harry feel that Hogwarts is no longer the safe haven it once was. So, he leaves, thinking that he can find a better place where he doesn't have to deal with Snape's negative presence.
In some fanfictions, it might be because Snape exposed a secret of Harry's that led to his departure.
One possible reason in fanfiction could be that Snape's constant bullying and unfair treatment finally pushed Harry over the edge. Harry might have felt that he couldn't take it anymore, especially if Snape made his life at Hogwarts unbearable in ways that the other teachers didn't notice or couldn't stop.
One possible reason could be that he feels he has fulfilled his purpose there. Maybe he has achieved certain goals related to protecting Harry Potter secretly, and he decides it's time to move on to other things. For example, in some fanfictions, he might leave to explore his own family history or to further his research on the Dark Arts in a more secluded place.
In fanfiction, one reason could be his desire for a fresh start away from the memories at Hogwarts. Maybe he feels he has done all he can there and wants to explore new magical knowledge elsewhere.
In some fanfictions, Harry might defend Snape because he comes to understand Snape's complex past. Snape was in love with Harry's mother, Lily, and his actions, though often harsh towards Harry, were sometimes motivated by his love for her and his role as a double agent against Voldemort. Harry, being a character who grows in understanding, could defend Snape once he learns these truths.
Snape terrifies Harry in fanfiction perhaps because of his strict and often cold demeanor. In the original series, Snape was harsh towards Harry, always seemingly looking for reasons to deduct points or give him detention. In fanfiction, this could be exaggerated. His dark appearance, with his long black robes and sallow complexion, also adds to the menacing aura.
One possible reason could be that he feels overwhelmed by the constant danger at Hogwarts. In some fanfictions, the threats he faces, like from dark wizards or dark magic within the school, might push him to the point where he thinks leaving is the only way to keep himself and his friends safe.
In fanfiction, Snape may treat Harry differently due to the power Harry possesses. Harry is a powerful wizard, and Snape might be afraid of what Harry could become. This fear could manifest as different treatment. Maybe Snape is trying to mold Harry in a certain way so that he doesn't go down a dark path, or perhaps he's just jealous of Harry's natural talent and tries to put him down as a result.
One reason could be that Harry Potter is the main character of the series. His story arc is central, and Snape is often a complex figure in relation to Harry. Writers might be more interested in exploring how Harry's actions and growth impact Snape's character and vice versa. Also, Harry has a broader range of relationships within the wizarding world that can be intertwined with Snape's story in fanfiction.
One reason could be to search for a powerful magical artifact outside of Hogwarts that they believe is crucial for defeating a new dark threat. In fanfiction, they might feel that Hogwarts has become too dangerous or restricted in terms of resources for their mission.