For Holmes, fanfiction might explore his past more deeply. For instance, his childhood traumas or his early days of learning detective skills. It could show how these experiences shaped the character we see in the movie. With Watson, his military background could be further delved into. Maybe there are some unresolved issues from his time in the war that come back to haunt him during the events of the fanfiction.
Moriarty in fanfiction often becomes even more complex. Writers might give him a more sympathetic backstory, like he was once a good person but was corrupted by society. This would add new layers to his character. Irene Adler could be made more independent and powerful. Instead of just being a love interest or a side character, she could be a full - fledged adversary or ally to Holmes, with her own motives and goals that are not solely related to Holmes.
Mycroft can be expanded in terms of his personal life. In the movie, we mainly see him in his official capacity. Fanfiction could show his hobbies, his relationships outside of work, and how he balances his duties to the state with his own desires. This would make him a more rounded and relatable character.