He might have a different view of magic. Growing up as Voldemort's son, he could be exposed to darker magic earlier. But instead of being corrupted by it, he could try to understand it in order to defeat it from within. This would give him a more nuanced understanding of magic compared to the original Harry Potter who was more on the side of 'light' magic from the start.
Harry would likely be more isolated. Other wizards might fear him or be suspicious of him. His relationships with friends like Ron and Hermione would be strained as they too would have doubts. But this isolation could also make him stronger and more self - reliant in some ways. He'd have to really dig deep to find his true identity, separate from the shadow of his father.
He would probably be more conflicted. His moral compass would be constantly tested. He might have a natural inclination towards power, like his father, but also have the good heart he developed growing up. So he'd be constantly fighting within himself.