The main characters are clearly young Joseph and Dio. Joseph is a character with a strong will and unique abilities, especially his Hamon. Dio, on the other hand, is a powerful vampire with his own set of formidable powers.
One popular plot is the discovery of a common enemy that forces Superman and Homelander to team up, despite their differences. Another is a battle for supremacy in a world where both exist, with different factions taking sides. And then there's the plot where Superman tries to reform Homelander, seeing the potential for good in him.
Well, it depends on how the fanfiction is written. If it's a more 'grounded' version where Homelander's public image and mind - games play a big role, he could potentially gain an upper hand. But generally, Superman has more raw power. He can move planets, and his speed is almost limitless. Homelander is strong and has flight and heat vision too, but Superman just seems to be on a whole other level in terms of pure power.
Perhaps a plot point where the power dynamic between them shifts unexpectedly. Maybe Joseph discovers a weakness of Dio that no one else knew, leading to a very tense and exciting confrontation. In fanfiction, this could be something completely new and creative that the author invents.