Both translations are prose, but their registers sit far apart. Murray writes in a formal, slightly archaic English that keeps classical distance: "The wrath sing, goddess, of Peleus' son" inverts normal word order, and phrases like "thus the plan of Zeus came to fulfillment" feel ceremonial rather than spoken. Rieu writes in plain mid-twentieth-century English: "Anger — sing, goddess, the anger of Achilles" opens with a short punch, and his narrative moves at a conversational pace. The difference sharpens in Book 21, where Rieu gives Achilles "Why make such a song about it?" and Murray gives "why lamentest thou thus?" Rieu's Achilles sounds like someone who means it. Murray's sounds like someone reciting. Murray stays close to the Greek word order and keeps Homeric epithets intact: "flashing-eyed Athene," "silver-footed Thetis," "single-hooved horses." This preserves the texture of oral formulaic composition and gives readers a sense of how Homer actually repeats and builds. Rieu cuts or rephrases those formulas to keep the prose moving. In Book 5, he drops Athena's speech-introduction entirely and starts with her direct words, while Murray retains "And the goddess, flashing-eyed Athene, answered him, saying." Murray's approach rewards readers who want proximity to the original; Rieu's rewards readers who want the story to pull them forward. Each removes something the other keeps.
Anger — sing, goddess, the anger of Achilles son of Peleus, that accursed anger, which brought the Greeks endless sufferings and sent the mighty souls of many warriors to Hades, leaving their bodies as carrion for the dogs and a feast for the birds; and Zeus' purpose was fulfilled. It all began when Agamemnon lord of men and godlike Achilles quarrelled and parted.
The wrath sing, goddess, of Peleus' son, Achilles, that destructive wrath which brought countless woes upon the Achaeans, and sent forth to Hades many valiant souls of heroes, and made them themselves spoil for dogs and every bird; thus the plan of Zeus came to fulfillment, from the time when first they parted in strife Atreus' son, king of men, and brilliant Achilles.