Because this question is broad, I will focus on one aspect of storytelling: voice.
Firstly, it is important to talk about what we mean by "traditional" and "modern." The modern period of literature is generally marked as beginning in the 1920s. However, there is something to Virginia Woolf's statement that the shift to modernity began in 1910, for early examples of Modernist literature emerged before the twenties.
Two such examples of "modern" short story collections include Edgar Lee Masters's Spoon River Anthology (1916) and Sherwood Anderson's Winesburg, Ohio (1919). Both collections prioritize the use of a first-person narrative voice, which became more common in Modernist literature. The subject matter of the stories in both collections is personal and illuminates less savory aspects of humanity through characters who seem real. Conversely, the Naturalist literature that had preceded this had used environment and physical frailties to depict character. Masters's work also helped to popularize the use of free verse, as the narratives are not exactly stories but rather poems that tell stories. Another aspect of modern literature is the blending of forms.
Later Modernist literature would also incorporate stream-of-consciousness and free-indirect discourse. The stream-of-consciousness device is particularly evident in the works of Virginia Woolf and James Joyce. In American literature, it was also used by William Faulkner.
Stream-of-consciousness narratives incorporate the thoughts of the characters in the narrative without the disruption of a narrator who lets us know that there has been a shift from action to thought. The narrative may also jump from the mind of one character and into another without any prior warning. In the nineteenth century, the use of voice was very tightly controlled and the narrative voice—usually a third-person omniscient narrator—was intrusive and sometimes told the reader what to think about a particular character.
Free-indirect discourse is a literary device that occurs in prose narratives written in the third-person omniscient voice. In the narrative, there is a sudden merging between the narrator’s voice, which is otherwise distant and observant, and that of the character who is speaking or having a thought. The narrator suddenly takes on the voice of the person or persons speaking, instituting their dialects or speech patterns. This technique may cause a bit of confusion for an inattentive reader, for the change can occur as soon as the next paragraph; in the following paragraph, the narrator will restore the traditional third-person voice. Zora Neale Hurston is an example of an author who used this device, particularly in her best-known novel, Their Eyes Were Watching God.