Lombardo's translation is more blunt and straightforward while Rieu's translation is more poetic and literary.
Sing, Goddess, Achilles' rage,
Black and murderous, that cost the Greeks
Incalculable pain, pitched countless souls
Of heroes into Hades' dark,
And left their bodies to rot as feasts
For dogs and birds, as Zeus' will was done.
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.