Perhaps Snape made a promise to Lily and then broke it. For example, he promised to stay away from certain dangerous people or activities, but he didn't keep his word. Lily, feeling betrayed, sends the Howler to express her extreme disappointment and anger.
It might be because Snape said or did something hurtful to Lily's friends. Lily is a loyal person, and if Snape insulted or harmed her friends in some way, she would be furious enough to send a Howler. In the context of fanfiction, there could be a new character introduced who Snape treats badly, and this angers Lily.
Hermione could be the sender. If Snape has made some unfair judgment in class against Gryffindor students, Hermione, with her strong sense of justice, might send a howler to him to voice her protest.
It could also be due to some external factors. Maybe there are threats from dark forces that he doesn't want to involve Lily in. If he is getting deeper into the dark arts or some dangerous situations, he might choose to give up on her to protect her from the potential harm that could come her way because of his associations.
One reason could be jealousy. Maybe Snape gets jealous of Lily's other relationships or her popularity at Hogwarts. Another could be a fundamental difference in their beliefs or values that grows over time into hatred.
It could be that in fanfiction, Snape has been through some additional trauma or experiences that change his perception of Lily. For instance, if he discovers something about Lily that he doesn't like, such as her involvement in some secret or her having a different side to her character that he finds unappealing. This would cause him to dislike her, as opposed to his more nuanced feelings in the original story.
One reason could be jealousy. If in the fanfiction Snape is getting more attention or praise from the wizarding world than Lily, she might start to dislike him. Another reason could be that Snape has a different view on blood purity, and Lily strongly disagrees with him, which leads to her hatred.
Maybe Snape did something that went against Lily's expectations or values, and she uses the howler to express her disappointment. For instance, he might have joined a group she doesn't approve of.
He might be initially shocked. After all, getting a Howler is a very public and embarrassing thing. He could be so ashamed that he immediately hides away to read it privately, his face turning red with a mix of anger and humiliation.
One reason could be jealousy. Snape might be jealous of James' popularity and charm. James, on the other hand, could be jealous of Snape's close friendship with Lily in their early years. Another reason is their different personalities. James was outgoing and bold, while Snape was more introverted and brooding. Lily might be attracted to different aspects of each of them, which could lead to the fight as they both compete for her attention.
His mother could send the howler. In the wizarding world, mothers can be quite fierce when it comes to family reputation and she might be unhappy with Sirius's actions or friends.
Perhaps in the fanfiction, James is seen as more accepting of Lily's friends and her social circle. Snape, on the other hand, might be associated with a darker crowd or have some beliefs that Lily doesn't agree with. For example, if Snape is too involved with the Dark Arts in the fanfiction, it could be a turn - off for Lily.