The exposition in Edgar Allan Poe's short story "The Cask Of Amontillado" relies a lot on the conjecture, surmise and guesswork of we the readers, so it is not a total exposition. We must presume by guesswork and clues that Edgar Allen Poe gives us, that there have been some aspersions cast on the narrators character, reputuation or expertise - that this has happened many (thousands?) of times on previous meetings and that the resulting offence taken has built into a volcano of hatred. From there we are taken in a slow and suspense-ful way to the denoument. By clues and guesswork we are taken to our own conclusion that the murder victim is hated and will be punished, and then executed. One foreshadowing clue is the warning about the air in the cellars.
What makes "The Cask of Amontillado" so intriguing is the gothic horror that is created by the unreliable narrator who, among many aberrrant behaviors, never explains to the reader what "the thousand injuries" of Fortunato are. So, there is no true "Initial Incident" in the exposition of Poe's narrative. Therefore, too, the exposition is incomplete since, by definition, it is "information essential to understanding the work."
The exposition is also part of the rising action which the initial incident is not. This rising action occurs after Montresor flatters Fortunato and induces the proud connosieur to examine his new cask of Amontillado. For, from this action the complication and turning points and denouement follow.
In a story, the exposition is the part that introduces the setting and the characters. It kind of tells us what is going on in general in the story.
To me the exposition in this story is the first four paragraphs. In this part, we find out that Montresor hates Fortunato, that it is carnival time, etc.
The initial incident is the thing that sets the whole story in motion. It is the start of the rising action, you could say. To me, the initial incident in this story is when Montresor meets Fortunato at the carnival.