In novels, the Prince of Demons usually did not directly PK the Princess because the Princess was usually a character in a novel or anime that represented a specific country or power, while the Prince of Demons was another character that represented a different power or goal.
In some novels or anime, there might be a battle between a demon prince and a princess, but it was usually because of a conflict of interest or goals between the two parties, not because the princess herself had evil characteristics. Therefore, the plot of the Demon Prince vs. Princess might appear in specific novels or anime.
There could be differences in details and expressions. The manga panel might show more nuanced emotions or fine details that the anime might not fully capture.
Guts is purely fictional. It's a concept or character that exists within the realm of imagination and creative storytelling rather than being based on reality.
Perhaps in this fanfic, Griffith has turned into something truly evil and a threat to the world Guts cares about. Guts, being the warrior he is, might feel it's his duty to put an end to Griffith to save others. Also, Guts could be driven by his own sense of justice. If Griffith has wronged innocent people, Guts may decide that killing him is the only way to set things right.