The change in his character could be significant. He would likely be more powerful magically, but also more dangerous. His upbringing with Snape and Bellatrix would shape him into a more cynical character. He might not have the same innocent view of the wizarding world as the original Harry. Instead, he would see the political and power struggles more clearly. His relationships with characters like Hermione and Ron would also be different. He might be more guarded around them, and they might be more hesitant to trust him due to his new parentage.
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.
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 '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.
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.
He might change in terms of his magical abilities. Maybe the Malfoy family focuses on different types of magic than what Harry was exposed to in the original story. So, he could be more proficient in dark arts - adjacent magic, but also struggle with the moral implications of using such magic. He may have to reevaluate his own identity and what it means to be a 'good' wizard.
The whole dynamic of the fight against Voldemort would shift. If Harry were Snape's son, Snape's loyalty and actions throughout the series would be re - evaluated. Harry might have access to different knowledge and skills that Snape could pass on. His connection to the Dark Arts might be more nuanced. Also, the way the other Hogwarts professors treat Harry would be different. They might be more cautious or more expectant of him, given his new family connection.
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.
He becomes more conflicted. Growing up as Lucius Malfoy's son, he is exposed to different values. So, instead of the Gryffindor - like straightforwardness, he may have a more calculating side.
He might become more in tune with nature. Since the Phoenix is a symbol of rebirth and nature in the wizarding world, Harry could develop a stronger sense of environmental magic. For example, he could be able to communicate with other magical creatures more easily, not just the Phoenix.
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.
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.