Well, Harry could be more confident in his own power. In the normal series, he had doubts about his abilities at times. But in this scenario, if he's set on destroying Snape, he has to be certain of his magic. He would be more decisive, not hesitating to use his skills against Snape. Also, his relationships with other characters might change. Some might see him as too aggressive in his pursuit of Snape, while others might support him wholeheartedly, creating a divide among his friends.
In this kind of fanfiction, Harry often becomes more proficient in potions and other areas that Snape is good at. His relationship with Snape also makes him more empathetic towards Snape's past and the Slytherin house in general. He may not be as quick to judge people based on their house. Also, he might develop a more complex view of the wizarding world's politics, as Snape is deeply involved in some of the darker aspects of it.
In some fanfictions, Snape might use his advanced knowledge of potions to create a special concoction to heal Harry. He could draw on his skills in Herbology as well to find rare ingredients for the potion.
Harry in this fanfiction might be a lot more complex. His identity is no longer that of the Boy Who Lived in the traditional sense. He has to come to terms with the fact that his parents are two very different and controversial figures. He may become more of an outcast, even among his friends at Hogwarts. His sense of right and wrong could be more blurred, as he has the influence of Bellatrix's dark ways and Snape's own morally ambiguous actions. He might be constantly questioning his own nature and whether he is destined to follow a path of good or evil.
Snape's character often develops from a cold and aloof figure to a more caring and father - like one. He starts to see Harry not just as James Potter's son but as an individual he is responsible for. His protective instincts grow, and he becomes more patient. For example, he might initially snap at Harry for small mistakes but over time, he learns to guide him gently.
His loyalty becomes more evident. In the original story, his loyalty to Dumbledore was somewhat hidden under his grumpy exterior. In this fanfiction, as Harry is learning to trust him, Snape's loyalty to the side of good and his care for Harry become more pronounced. He might take more risks to protect Harry and Hogwarts, and his actions are no longer so easily misinterpreted as malevolent.
In these fanfictions, Snape's character often develops from a seemingly cold and unfeeling person to someone with more depth. When he saves Harry, it starts to chip away at his facade. For example, he might initially save Harry out of obligation but as time goes on, he realizes he has a genuine care for the boy. This is different from his canon behavior where his feelings are mostly hidden.
Another way Snape could develop is by forming unexpected alliances during the events of the Goblet of Fire in fanfiction. He might team up with other teachers or even some of the students in a way that forces him to step out of his usual solitary and grouchy self. This new experience of working with others could make him more open - minded and change his overall character for the better. He could learn to trust others more, especially if they all have a common goal related to the strange events happening during the tournament.
Snape's character could develop in a way that he becomes more confident in time - travel scenarios. He starts using his knowledge of the future to his advantage. For instance, he could use potions knowledge from the future to create new spells or remedies in the past, which boosts his standing among the Hogwarts faculty and changes how others view him. This new - found confidence might also make him more open about his feelings, especially towards Lily.
It could be Bellatrix Lestrange. She was always a fanatic follower of Voldemort and hated those who she thought were against the Dark Lord. Since Snape was a double - agent, she might have discovered his true loyalties and decided to kill him in the fanfic.