The very fact that Bradbury's story takes its title from Teasdale's poem serves as a powerful confirmation that the story is very much in conversation with it. Indeed, not only does Bradbury reference Teasdale's work with the story's title, he also quotes it within the text, having the house recite the poem directly.
Both poem and story contain similar thematic material, as they both involve the possible destruction of humanity. Teasdale's poem concerns the existential insignificance of the human species: if we were to go extinct, the world would go on, largely unaffected by that extinction. In this respect, however, one might suggest that Bradbury's story is quite different in its tone and imagery. After all, Teasdale's poem presents a world filled with beauty and flourishing with life, whereas Bradbury's picture is much more violent and desolate, shaped by the imagery of a city reduced to nuclear devastation. Indeed, note, too, that nature itself is presented more violently in Bradbury's story, given the imagery of the fire as it consumes the house while the house fights a last desperate struggle to survive. In this respect, one might say that in both works, nature ultimately conquers humanity and is expressed as the more enduring and powerful force, but Bradbury's story is marked by a violence and brutality, not just reflected in the destruction of humanity by nuclear war but also within his depiction of nature itself.
In this respect, the two works mirror one another, sharing similar themes and mindsets, yet the two are not in complete agreement, either in their aesthetics or their thematic content. While Teasdale's poem contains a darkness beneath its idyllic imagery (it ultimately concerns the subject of human extinction after all), Bradbury takes that darkness and brings it to the forefront, as seen in the destruction of the city and the destruction of the house that survived it.