It is important to realise that the "liquor" that the speaker in Dickinson's poem confesses she indulges in to excess is not actually alcohol. Dickinson actually creates an extended metaphor in the "liquor never brewed," comparing it throughout to her joy in nature that is endless and she can never get enough off. Note how the speaker describes herself in the second stanza:
Inebriate of air am I,
And debauchee of dew,
Reeling, through endless summer days,
From inns of molten blue.
Here she uses words that we normally associate with drunkards, such as "Inebriate" and "debauchee," to highlight and emphasise the extreme joy she takes in nature.
In "The Drunkard," by contrast, the alcohol is definitely literal, and the reader is shown the way in which the speaker's father's alcoholism is a real burden for all the family and not a source of joy in any sense. Larry, the speaker, by accident becomes drunk and follows in his father's footsteps. However, although this is humorous, the reader is never allowed to forget the very serious and damaging nature of his father's alcoholism and how the whole family suffers. Mick has pawned all the furniture in the past to fuel his need for a drink, and the narrator, as an adult, looking back on his childhood, confesses that "I could never get over the lonesomeness of the kitchen without a clock." Whereas the tone of Dickinson's poem is one of unbridled joy, the tone of this short story is one of grim, humorous realism. Although in this short story, the situations are reversed, with Mick forced to look on and watch his son drunk, the reader is made to think of all the times that Mick would have returned home drunk with his wife and son watching on, helpless.