Both poems start with mimicry of Greek epic poetry by employing praise of the subject. Swift praises Corinna as the "pride of Drury Lane" (an upscale area frequented by impoverished women with nothing but themselves to sell). Pope praises Belinda by saying the subject of his poem is insignificant (i.e., "slight") "but not so the praise" of Belinda. Each poet quickly turns the poem to the intended topic.
Swift says "no shepherd sighs in vain" for Corinna's love, which is a satirical pastoral allusion meaning that Corinna is beloved of no one. Pope asks "what strange motive" could cause a "Goddess" like Belinda to turn her feelings to rage: "Could make a gentle Belle reject a Lord / ... / And in soft bosoms [dwell] such mighty Rage?" Both poets proceed to develop their satirical poems through detailed description, particularly description of the lady's toilette (i.e., dressing or undressing ritual).
It is in the description of the toilettes that Swift and Pope differ. Swift uses deep irony in his descriptions. His irony is calculated to make Corinna a tragically sympathetic character (“So bright a battered, strolling toast”) and turn his satire to a serious social point (“With pains of love tormented lies”). Pope uses light-hearted irony that is replete with Classical allusion (“Of airy Elves by moonlight shadows seen”). His irony is meant to make Belinda looking adoringly vain and turn his satire to a foolish feud between friends (“Beware of all, but most beware of Man!”).
In "A Beautiful Young Nymph Going to Bed," Swift uses what is usually thought of as an appealing ritual of undressing to reveal a young woman who is bereft of nearly everything--except life and pain--because of the practice of prostitution and its associated diseases:
With pains of love tormented lies;
Or if she chance to close her eyes,
Of Bridewell and the Compter dreams,
And feels the lash, and faintly screams;
(NOTE: Bridewell, a woman's prison. Compter, prisons controlled by sheriffs)
In The Rape of the Lock, Pope uses the ritual of dressing in the morning to show how pampered, privileged, and vain the lovely Belinda and her feud are:
What wonder then, fair nymph! thy hairs should feel,
The conqu'ring force of unresisted steel?
The satire of the one is meant to reveal the social injustice and evils of Corinna's tragic life, with the aim of bringing about a change in society, while the satire of the other is meant to reveal the triviality of a quarrel between friends, with the aim of bringing about a reconciliation between friends and families.