The similiarity between these two sections of the poem lies in the way that Pope creates a world where the normal world interacts with a kind of fairy world, with a number of attendant fairies that hover around people and try to influence what happens in their life. Thus it is that Pope at the beginning of the poem in Canto I reveals:
Know then, unnumber'd Spirits round thee fly,
The light Militia of the lower sky:
These, tho' unseen, are ever on the Wing,
Hang o'er the Box, and hover round the Ring.
Think what an Equipage thou hast in Air,
And view with scorn Two Pages and a Chair.
Thus Pope creates a secret hidden world of fairies and spirits that interacts with our own world and where the fairies are characters in themselves and take the part of various characters, trying to support and protect them. This theme and playful, whimsical tone is picked up in the section quote from Canto III that you have identified, which is just before the lock is actually taken. It describes how the attendant fairies try and prevent this from happening, blowing the hair back with their wings away from the "Foe" who is threatening to take it:
Swift to the Lock a thousand Sprights repair,
A thousand Wings, by turns, blow back the Hair;
And thrice they twitch'd the Diamond in her Ear,
Thrice she look'd back, and thrice the Foe drew near.
Just in that instant, anxious Ariel sought
The close Recesses of the Virgin's thought;
As on the Nosegay in her Breast reclin'd,
He watch'd th' Ideas rising in her Mind,
Sudden he view'd, in spite of all her Art,
An Earthly Lover lurking at her Heart.
Amaz'd, confus'd, he found his Power expir'd,
Resign'd to Fate, and with a Sigh retir'd.
The similiarity is based on the way that both extracts of this poem therefore feature this fairy world and explore how it interacts secretly with our own world, with the fairies taking sides and trying to influence the course of fate, but also how, as the second quotation makes clear, this is often a futile struggle.