Well, it really depends on the creativity of the fanfiction authors. In some Twilight fanfiction with 'Caveman Edward', he could be shown as being overly protective in a very primal way. For example, he might be more territorial about Bella, which is different from his more refined character in the original. It can be an interesting exploration of his character if done well.
Edward in Twilight fanfiction is typically depicted as having an intense internal struggle. His love for the human girl is always at odds with his vampire instincts. He may be more conflicted in fanfiction than in the original books, as authors explore different scenarios. For example, some fanfics might show him having to choose between his family and his love in a more extreme way. Also, his relationship with other vampires can be further developed in fanfiction, showing his loyalty or lack thereof in different situations.
One great fanfiction could be where Edward and Bella's relationship is explored more deeply in a different vampire society setting. For example, they might be in a vampire clan that has unique rules about human - vampire relationships.
In many Twilight fanfictions, Edward and Bella's kiss is often depicted as an extremely passionate and intense moment. Some fanfictions might explore different scenarios around this kiss. For example, it could be set in an alternate universe where they meet under different circumstances before the kiss. The kiss in fanfiction can also be used to show a deeper connection between them than what was shown in the original books. It allows the writers to play with the emotions and the relationship dynamics in a more creative way.
There's also 'Twilight Reimagined: Edward and Jacob'. In this one, the author completely reimagines the story. Edward and Jacob are forced to work together in a world where the Volturi are more powerful than ever. It has a lot of great character development and the plot twists keep you on your toes.
One great 'Twilight Edward Story Fanfiction' is 'A Different Shade of Twilight'. It offers a fresh take on Edward's character, exploring his internal struggles in a new and interesting way.
Another common theme is the exploration of Edward's vampiric nature. Fanfic writers might focus on how he deals with his thirst for blood while being in love with a human, or how his vampiric abilities influence his relationships.
The character interactions are also unique. You get to see how the Twilight characters like Bella or Edward Cullen might interact with a figure like King Edward. Maybe they would be in awe of his royalty or he might be intrigued by their supernatural abilities. It opens up a whole new world of relationships to explore in fanfiction.
A common theme is the 'what - ifs'. What if Bella had chosen Jacob instead? Fanfictions often explore this alternative storyline and how it would affect the vampire - human relations in Forks. It can be really interesting to see how the characters develop in this different scenario.