He'd probably be more confident in his water - related abilities from an early age. Since he's raised by Poseidon, he might have a deeper understanding of the sea and its power.
His relationship with authority figures other than Poseidon would be more strained. He's used to being under Poseidon's guidance, which is different from the more structured rules of the demigod camps. So he might be more rebellious towards those in charge at the camps, as he's used to his father's more fluid way of doing things.
He'd be more in tune with the sea. He might be calmer, like the ebb and flow of the ocean. His reactions to things would be more measured as he's used to the slow, powerful nature of the sea. For example, instead of getting angry quickly, he'd wait and think like the sea does before a big wave hits.
Percy's powers would be more complex. He could have the ability to communicate with sea deities in a different way. Maybe he could use Hera's diplomatic skills to make deals with them or resolve disputes. His control over water could also be extended to other substances that have a connection to water, like steam or ice, in a more efficient way than before. And his powers of persuasion, which could be influenced by Hera, might be used to calm angry sea storms or make sea monsters cooperate with him, which is an interesting combination of his Poseidon - given powers and Hera - influenced skills.
In a fanfiction where Percy is the son of Hades and Poseidon, his powers would be truly extraordinary. He could have the ability to create portals in water that lead to the Underworld. This would give him an advantage in both battle and travel. His hydrokinesis could be tainted with the essence of death, so when he attacks with water, it could cause a draining effect on his opponents, sapping their life force. Additionally, he might be able to sense the presence of the dead in any water source, which could be used for both reconnaissance and protection.
His character could change in several ways. He would have a new sense of identity. No longer just a demigod with powers, but now something more 'aquatic'. He might be more curious about his father's world and want to spend more time there. This could lead to some conflict with his human side, as he has to balance his life on land and his new underwater existence. Also, he could become more of a leader among the sea creatures, as he has a unique connection to them through the tail.
In this fanfiction scenario, Poseidon's relationship with Percy would be full of twists and turns. At first, Poseidon might be shocked and a bit angry. How could his son be related to Leto? But as time goes on, he realizes that Percy is still his son and that this new connection could be used to their advantage. He starts to work with Percy more closely, guiding him on how to balance his powers from both Poseidon and Leto. They might have some father - son talks about the new responsibilities Percy has and how he can use his dual heritage to protect the world better. Poseidon could also start to see Percy as more of an equal, as he is now part of a more complex and powerful family tree.
Percy would likely become more confident in his abilities. Apollo's influence would make him more outgoing and perhaps a bit more self - assured. He'd also probably have a greater appreciation for beauty and the finer things in life.
Well, if Percy was more powerful, he could act as a stronger mediator between Zeus and Poseidon. He might be able to force them to face their issues more directly. For example, he could use his increased power to create situations where they have to work together, like in the face of a new and extremely powerful enemy. This could lead to a begrudging respect between the two gods and a shift in their relationship from one of constant bickering to a more cooperative one, at least when Percy is around.
His character would likely be a blend of Athena's wisdom and Poseidon's boldness. He'd be more calculating in his actions, thinking several steps ahead like Athena, but also not afraid to take risks when necessary, similar to Poseidon. He might have a natural inclination towards both strategy and brute force, which could make him a very unpredictable and powerful character.
It can also make Percy more empathetic. Since he has such a powerful and protective father, he realizes the importance of protecting others too. He may be more likely to step in and help those in need, just as Poseidon does for him. This protection also gives Percy a sense of security that allows him to explore his own identity and powers more freely. He doesn't have to constantly worry about being alone or unprotected, so he can focus on growing as a hero and as a person.
Percy would likely become more confident in his powers. As Apollo's brother, he might gain access to some of Apollo's knowledge, which could make him a better strategist. For example, he could use Apollo's understanding of the sun's patterns to his advantage in battles.