Both translations are in verse, but they handle line length and register very differently. Mitchell writes in loose iambic lines that stay close to contemporary American speech. His opening, "The rage of Achilles, sing it now, goddess, sing through me," is direct and conversational; his Achilles in Book 21 says simply "you too must die. Why all this moaning about it?" Green writes longer, more crowded lines that carry more of the Greek's information per line. His version of the same moment reads "So, friend, you too must die: why then lament thus?" The word "lament" and the syntax "why then lament thus" sit at a slightly higher register than Mitchell's phrasing, and Green's lines throughout carry more formal weight. Green's approach stays close to the Greek's word order and content, keeping proper names in transliterated forms with diacritical marks (Pēleus, Hādēs, Arēs) and retaining lines Mitchell cuts. In Book 9, Mitchell ends Achilles' speech after "my life will be long and peaceful," dropping Zeus's extended hand and the emboldened troops entirely. Green keeps both lines. Green also preserves the Greek's repetition in Book 6 ("as the generation of leaves, so is that of mankind"), where Mitchell paraphrases into a four-line expansion. Mitchell removes details for forward momentum; Green keeps them at the cost of some syntactic awkwardness, particularly in lines like "two contrary spirits go with me until the end that's death."
The rage of Achilles—sing it now, goddess, sing through me
the deadly rage that caused the Achaeans such grief
and hurled down to Hades the souls of so many fighters,
leaving their naked flesh to be eaten by dogs
and carrion birds, as the will of Zeus was accomplished.
Begin at the time when bitter words first divided
that king of men, Agamemnon, and godlike Achilles.
Wrath, goddess, sing of Achilles Pēleus's son's
calamitous wrath, which hit the Achaians with countless ills—
many the valiant souls it saw off down to Hādēs,
souls of heroes, their selves1 left as carrion for dogs
and all birds of prey, and the plan of Zeus was fulfilled
from the first moment those two men parted in fury,
Atreus's son, king of men, and the godlike Achilles.