"The Case of the Colonel's Suicide," by Rafe McGregor, in Sherlock Holmes Mystery Magazine, Issue 29, 2022.
A pastiche is a story written in the style and characters of another author, attempting to add to the existing corpus of those works. Fan fiction, if you like.
A homage, on the other hand, uses elements of the original author's universe in a new way.
You could call this story a homage since it name-checks two characters by a well-known author. But that would be a stretch, because the tale would work just as well if those two names were replaced by Smith and Jones.
We are in Victorian England and the narrator is Chief Inspector Langham of the Metropolitan Police. A retired colonel has committed suicide and Langham, himself a former military man, is asked to examine the scene.
What he sees convinces him that the colonel, although burdened with debts and other serious problems, was murdered. And so Langham begins to delve deeper into the man's troubled past.
What makes this story stand out for me is its use of detail. My knowledge of 19th century English military customs is nil, but I am convinced that McGregor knows his stuff, and he makes it fascinating.