He'd probably be more confident. Dante is a very confident character, and if Harry is his son, he might inherit that trait. He'd be less likely to be intimidated by bullies like Draco or even the Death Eaters.
His view of good and evil might change. Growing up as the son of Dante Sparda, he'd be exposed to the idea that there are different levels of evil and different ways to fight it. In the wizarding world, the divide between good and evil seems more straightforward. But with his new heritage, he'd understand that there are gray areas. This could make him more empathetic towards some of the characters who are on the 'wrong' side, like some of the Slytherins who might be pressured by their families to side with Voldemort.
Well, it could involve Harry having some of Dante's devil - slaying abilities. Maybe he uses them to fight the dark forces in the wizarding world that Voldemort represents in a more unique way than just magic. For example, he could have enhanced speed and strength like Dante to take on Death Eaters.
Harry's character could change in that he becomes more of a leader. With the special abilities and heritage from Hecate, he may find himself in situations where he has to take charge. He might also develop a deeper sense of responsibility, not just for his friends but for the wizarding world as a whole. He realizes he has a different role to play now that he is the son of a goddess. Also, he could become more attuned to the more mysterious and darker aspects of magic, which might make him seem a bit more brooding at times.
He becomes more conflicted. Knowing he's Voldemort's son, he's constantly in turmoil about his true nature. His confidence might waver as he wonders if he's capable of the same evil as his father.
Well, it could be that Harry Potter discovers he has a connection to Sparda. Maybe he has some latent power that is related to Sparda's abilities. He then has to learn to control this new power while still dealing with the threats in his own wizarding world. And perhaps he uses his new - found powers to fight off dark wizards in a more badass way.
His character would become more complex. He'd be more confident in his abilities, not just relying on his magic but also on his Sparda - given powers. He might be a bit more of a loner, as he tries to understand his unique identity. Harry would also be more fearless, knowing he has the strength of Sparda in him. This could lead to him taking more risks in his battles against the dark arts.
His character becomes more independent. As the son of Artemis, he has a different kind of power and heritage that sets him apart from other wizards. This could lead to him relying less on the help of others and more on his own instincts and abilities. He might also be more secretive about his new powers, which adds a new layer of mystery to his character. For example, he could start to disappear at night to train in the moonlight without telling his friends.
Harry would probably become more physically bold. Since Hercules was known for his strength and heroic deeds, Harry could be more likely to take on challenges head - on, using his physical abilities more often. His personality might also change in terms of his leadership. With the added boost of being related to a great hero, he could be more assertive in leading his friends and the Order of the Phoenix. He'd no longer be just the Boy Who Lived, but also the son of a powerful demigod, which would surely impact how he views himself and his role in the wizarding world.
He'd probably be more morally ambiguous. Growing up as Crowley's son, he might not have the same clear - cut sense of right and wrong as the original Harry. He could be more cunning and less hesitant to use underhanded tactics if necessary.
In this fanfiction, Harry's change is quite significant. Firstly, his view of himself and his place in the world is completely altered. Instead of just being a wizard, he is now part of something much bigger. This new identity gives him a different kind of courage. He's not just fighting for the wizarding world against Voldemort, but also has a connection to Asgard. His relationships with his friends also change. They might see him in a new light, and he might have to deal with their different reactions. He could become more of a leader, using his new powers and status to inspire others. And his magic might even be influenced by his Asgardian heritage, making it more powerful or having different manifestations.
In 'Harry Potter son of Loki' fanfiction, Harry changes significantly. His sense of identity is greatly shaken. He no longer just sees himself as a simple wizard. He now has this connection to Norse mythology. This makes him more curious about different forms of magic and otherworldly things. He may become more of an outcast as well, since his heritage is so different. But this also makes him stronger in some ways. He has to rely on himself more and develop his own unique set of skills and powers that come from both his Potter and Loki lineage.