He'd likely be less jaded. At Hogwarts, he faced a lot of hardships like being bullied by the Marauders. Without that, he might be more open and less defensive. His social skills could be better as he wouldn't have developed the same level of distrust towards others.
His knowledge of magic would be different. Hogwarts has a set curriculum, but without it, he could focus more on areas he's truly interested in from the start. For example, he might be even more of an expert in potion - making earlier on, as he wouldn't have to spend time on other Hogwarts - required subjects. Also, his view of the wizarding world might be more positive as he wouldn't be exposed to the same in - house rivalries and politics at Hogwarts.
Snape's character would lack the strong connection to Hogwarts that defined him in the original story. Without the influence of Hogwarts' professors and the house system, his sense of identity would be different. He may not have the same internal conflict between his loyalty to the Dark Lord and his love for Lily, as the events that led to this conflict at Hogwarts would not have occurred. Instead, he could form different loyalties based on his new experiences outside of Hogwarts, perhaps to a different magical community or a group of independent wizards.
One possibility could be that Snape grows up in a different magical environment, perhaps in a small, hidden magical community that has its own unique way of teaching magic. He might develop his skills in potion - making and the Dark Arts in a more self - taught or community - guided way. Without the influence of Hogwarts, his relationships would be very different. He may not have the same intense rivalries or unrequited love for Lily Potter.
Snape would likely become more confident in his place at Hogwarts. As the heir, he has a new - found sense of belonging that could soften his harsh exterior. He might be more open to helping students, especially those who show an interest in the history of Hogwarts.
Well, it would be a lot less antagonistic. Without the whole Death Eater past and the spying for Dumbledore while seeming to be on Voldemort's side, Snape wouldn't have that built - up resentment towards Harry. He might actually see Harry as a talented young wizard worthy of proper mentorship, instead of constantly being so harsh on him.
Her relationships would be different. Instead of having school friends like Harry and Ron, she might form alliances with other self - taught wizards or those from different magical backgrounds. This could lead to a more diverse and complex social network, which in turn would shape her character in unique ways. For instance, she could become more diplomatic as she deals with different magical cultures outside of the Hogwarts - centric wizarding world.
Hermione and Ron might not become friends in the same way. Since Harry was the catalyst for their friendship at Hogwarts, without him there, they might not cross paths as easily. Hermione could be more focused on her studies alone in a different school, and Ron might be stuck in his family's shadow without Harry to bring out his better qualities.
Snape's character could become more complex. He might start to incorporate some of Merlin's wisdom into his own actions. For example, Merlin was known for his ability to see the good in people, and Snape could start to do the same. Instead of just being bitter and resentful towards Harry, he could start to see Harry as a potential ally in a greater fight, much like Merlin saw the potential in the young knights of Camelot.
As a wolf Animagus, Snape might be more in touch with his primal instincts. His usual guarded and calculating personality could be tempered by the wild nature of the wolf. He may be more impulsive in his actions, acting on instinct rather than over - thinking every move like he does in his human form.
One way could be through a magical oath. In some fanfictions, Snape might have sworn an oath to protect Hogwarts at all costs, which binds him to the school. Another way is through his love for Lily Potter. Since Hogwarts was a place that was important to her too, his emotional connection to her could translate into a binding to the school. Also, he could be bound by the magic of the place itself. Hogwarts is full of ancient magic, and it might recognize Snape's loyalty and determination, thus binding him to it.
Well, first of all, their friendship would probably make the staff room a more harmonious place. Professors would see that two people from such different backgrounds and houses can be friends. In the classrooms, Snape might be less harsh on Gryffindor students because of his friendship with Lilly. And Lilly could use her friendship with Snape to encourage more Slytherin students to be open - minded about Muggle - borns. This would change the overall atmosphere of Hogwarts, making it a place where friendship and acceptance are more valued than house rivalries.
Snape's character could become more calculating and cold - hearted. Tom Riddle is not exactly a nurturing figure, and being raised by him would expose Snape to a world of power - seeking and self - interest. Snape may lose some of his capacity for love and affection that he had for Lily in the original story. He would be more focused on achieving power and status within Riddle's regime. His relationships with other characters would be more transactional. For instance, he would not form the same kind of complex relationship with Narcissa Malfoy as he did in the original story because his priorities would be different, being centered around Riddle's goals rather than his own personal loyalties and emotions.