"Prufrock" and "Do Not Go Gentle" share many themes. For example, as you point out, both are concerned with aging, the threat of death. On a more basic level, both deal with the problem of individuality, or of the meaning of an individual's life. Where they differ is in rhetorical stance, tone, and outcome.

In "Prufrock," the poem is spoken by a narrator who, in a kind of confession, expresses his extreme alienation from everyday life and his sense that life is slipping away uselessly. The narrator is paralyzed by fear: fear of his own death, but also fear of asserting himself, or of inserting himself into a life that he, as poet, merely observes.

When he famously asks if he "dares to eat a peach," the suggestion is that the decision of what to eat is simultaneously too mundane to mention without self-parody but also an expression of a real fear of how others will see him, or of how the minute actions one performs everyday determines who we are. The narrator is a kind of coward, but one who (through the writing of the poem) is trying to come to terms with his own cowardice.

In Thomas's poem, the tone is strikingly different. Here, the poet addresses his dying father. His words are active and direct. The poem lacks the ambivalence towards life we find in "Prufrock." Death, for Thomas, is personal, and his urging to fight against "the dying of the light" or death can be understood as both a kind of general principle (one must fight to preserve one's identity in the world) and a specific request ("Father, please don't die"). Nevertheless, both poems being are about the human condition, or the struggle to retain identity and self-knowledge even in the face of mortality.