The narrator in "The Black Cat" is an obvious sociopath, incapable of feeling true remorse or guilt. Nevertheless, as he tells his tale, he recounts a regression into further depths of depravity. Therefore, when he kills his first cat, Pluto, he is somewhat horrified at his deed, and for months it bothers him with a "half-sentiment that seemed, but was not, remorse." Still, as he reports it, he is not satisfied with himself after killing the cat, and he plunges further into alcoholism.
After killing his wife, however, no such half-sentiments plague him. Instead, he is intent on covering up his crime, and when he has carefully hidden the corpse behind a brick wall in the cellar, he is pleased with himself, stating, "I felt satisfied that all was right." He only resolves after that to kill the cat, and when he believes it has run away, he experiences great relief. He begins to believe that his future happiness is guaranteed and reports, "The guilt of my dark deed disturbed me but little." Strangely, he is actually proud of his cover-up, so much so that he draws the police who are investigating his wife's disappearance into the cellar to further confirm to himself his skill at obfuscating his crime. He feels absolutely no guilt at this point--only pride in successfully hiding the body.
The difference in the way the narrator reacts to the two killings, if we can trust his account, reflects his further descent into depravity to the point where he is not only not quite capable of feeling remorse for his evil deed, but in the end completely incapable of feeling guilt. However, keeping in mind that the narrator in the story is highly unreliable, one might question whether he actually feels any level of remorse over the death of the cat and is troubled by it only because his crime is apparent to his neighbors and his wife. When his later crime is fully hidden, on the other hand, he feels comfortable with it. Such reactions are consistent with sociopathic attitudes and behavior.