As with many Poe stories, both narrators are certifiably insane. However, the narrator of "The Tell-Tale Heart" starts his narrative by attempting to reassure his audience that he certainly is not crazy. As proof, we are asked to note "how calmly [he] can tell [us] the whole story"--words a sane person would never utter. He explains that he's just "nervous," and it comes with certain gifts, even! "Above all," he writes, "was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell." A man who hears things in heaven and earth and hell? He certainly is a sandwich short of a picnic.
In contrast, the narrator of "The Cask of Amontillado" seems quite sane in comparison. He's just angry. We--the audience--can even identify with him a bit. He's been snubbed and insulted by an aristocrat, and he's finally had enough and will get revenge.
The he narrator of "The Tale-Tell Heart" admits to loving his intended victim, though, while the narrator of the "Cask of Amontillado" hates his. In the former, he cannot explain at first why he even wants to kill the old man. He doesn't hate him, he doesn't want his gold, but he is driven--haunted, even--to kill him. Finally, he decides it must be the old man's eye that he cannot stand. In the latter, though, we're told up front that there are reasons: "THE thousand injuries of Fortunato I had borne as I best could, but when he ventured upon insult I vowed revenge." (Of course, considering that we eventually realize that we're dealing with a madman, we should call into question whether Fortunato ever actually did anything to him.)
Both narrators pride themselves on being overly pleasant to their intended victims until they could go through with their plans, so neither victim would suspect a thing.
Keep in mind that TTH madman's entire narrative is designed to convince his audience that he is not crazy (and has quite the opposite--creepy--effect). The COA narrator, on the other hand, is bragging. That's right. Bragging.
TTH madman chooses a most unorthodox way to murder his victim, and it is most time-consuming. It requires him to open the man's chamber door exactly at midnight every night (slowly as the minute hand of a clock) and just as slowly open the crevice of the lantern until a ray shone out; only on the eighth night did his plan pay off. COA madman, on the other hand, chooses a time-tested method: burying his victim alive; all he has to do is get him drunk and trick him into the catacombs of his own family home then lock him to a wall. If he says nothing--which he doesn't for fifty years--he'll never be found out.
TTH madman has to deal with the police, however, because the old man shrieked before his heart attack killed him. COA madman has no such bother. When the police arrived, TTH madman's own senses--particularly his acute sense of hearing (and possibly his bonafide lunacy)--betray him, despite his over-confidence (another trait the two share*). While he may be telling us the story "calmly" (at least in his opinion), he loses control when he hears the heart beating. He begins talking louder, pacing, and behaving like the lunatic he is. The police don't notice this; he thinks they're enjoying his hysteria, so he finally rips up the floorboards and admits his crime.
* Montresor's confidence is clear as he and Fortunato descend into the family crypt. Fortunato has a cough, and Montresor suggests a couple of times that they return to the surface, because he doesn't want him to catch a cold. He also tells Montresor what his family coat of arms is and its motto (essentially, "No one hurts me and gets way with it"), which might have been a warning to a man who had knowingly hurt him.
As far as we know, Montresor is never caught. We don't know who he's telling his story to, or why. Perhaps a grandson? The unnamed narrator of TTH, however, seems to be trying to convince someone that he isn't crazy, which leads me to believe that his audience is a lawyer or a judge.