Assuming that "importance" refers to literary importance—or the importance of the scene to literary elements and development—rather than referring to social criticism importance, then the literary importance of the Denham's Dentifrice commercial is that it quite intensely reveals the violent inner struggle Montag is going through. He is trying to extricate himself from one false society and embed himself in a true society because he has learned "of a time when books were legal and people did not live in fear" (Jepsen and Johnston, spaceagecity.com).
"Shut up, shut up, shut up!" [His] was a plea, a cry so terrible that Montag found himself on his feet, ... this man with the insane, gorged face, the gibbering, dry mouth, the flapping book in his fist.
Montag has been reading his stolen books to Mildred, whose only response is, "Books aren't people. You read and I look around, but there isn't anybody!" when an electronic dog comes sniffing at their front door, exhaling "the smell of blue electricity blowing under the locked door." Montag—the fireman—knows full well what the sniffing dog means. Beatty knows Montag has stolen and expects the return of the book ("If I pick a substitute and Beatty does know which book I stole, he'll guess we've an entire library here!") that very night. He is, as he says himself, "numb" ("I'm numb, he thought") as he slams the house door and goes to board the subway. He has decided to go to Faber and ask to have a duplicate of the stolen book made so he can safely—safely for himself and Mildred and safely for the book—return the stolen book to Beatty.
"There's only one thing to do," he said. "Some time before tonight when I give the book to Beatty, I've got to have a duplicate made."
Riding on the subway amongst so many people, Montag is both scared of what he is doing and earnestly determined to memorize a portion of the New Testament that he holds open (foolhardy action) in his hands. The Denham's Dentifrice jingle has all the passengers tapping their feet and quietly singing along with the jingly words. The jingle acts as a literary counterpoint as it lauds "Denham's Dentifrice. Denham's. Spelled D-E-N-" while Montag struggles to retain the sentence "Consider the lilies of the field. ... Consider the lilies, the lilies, the lilies ...." The old, meaningless society fights against Montag's mind, as detergent would against impurities, "Denham's dental detergent," until he breaks down, shouting "Shut up, shut up, shut up!" to the "rhythm of Denham's Dentifrice, Denham's Dandy Dental Detergent, Denham's Dentifrice Dentifrice Dentifrice, one two, one two three,...."
Montag battles, against dire consequences, for thought under the fear, strain, desperation and desire that compel him forward to Knoll View (symbolic as a rise from which to gain a vantage place for seeing the panorama). There he hopes to perpetuate the life of a book that will stand against the totality of "'the family'" and the "White Clown" and keep alive the society that he seeks to embrace, the free society where books were desired. Montag's struggle for one society over the other is the literary importance of this scene in "The Sieve and the Sand." Montag's efforts to memorize "Consider the lilies of the field" are so much sand through a sieve: "the sand was boiling, the sieve was empty. Seated there in the midst of July, without a sound, he felt the tears move down his cheeks."
"'Denham's. Spelled : D-E-N-'
They toil not, neither do they...
A fierce whisper of hot sand through empty sieve.
'Denham's does it!'
Consider the lilies, the lilies, the lilies...
'Denham's dental detergent.'"
Regarding social criticism, the social importance of the scene is significant also. After all, Bradbury wrote Fahrenheit 451
as a social dystopia novel because the new technology of television and the resultant escalations in advertising were disconcerting to Bradbury and cause for his negative prognostics of the future. As stated by Biography.com, Bradbury expressed his "distaste for television
" when he explained Fahrenheit 451
as a novel about "how television drives away interest in reading."
So, while "importance" relating to the Denham Dentifrice scene in "The Sieve and the Sand" is a significant factor of the novel as social criticism, the social importance is quite different from the literary importance. Reflecting on Bradbury's opinion of television and on the "detergent" characteristics of Denham's Dentifrice, we might arguably say that the social importance of this scene is that electronic entertainments, like television entertainment, including the jingles of advertisements (so popular on television and other modes of entertainment), scrub away the productive, intelligent and independent thoughts in a person's mind as though they were impurities, even as the detergent dentifrice, "Denham's Dandy Dental Detergent," scrubs away impurities on teeth.