Greek names change between translations because English has inherited more than one route from ancient Greek to the modern page.
A reader moves from one translation to another and suddenly Aias has become Ajax. Odysseus is Ulysses. Herakles appears as Heracles or Hercules. Akhilleus may be Achilles.
Usually this is not a disagreement about who the person is. It is a disagreement, or simply a convention, about how the name should travel.
One route is direct transliteration from Greek
Ancient Greek is written in a different alphabet. A translator or editor who wants the English spelling to stay visibly close to the Greek can transliterate the letters into the Latin alphabet.
That sounds mechanical until one asks what counts as “close.” Greek κ can appear as k in a deliberately Greek-looking transliteration, while older English classical convention often uses c. Greek χ can be represented conventionally as ch or more explicitly as kh. Long vowels and accents may be marked in scholarly work and omitted in a general edition.
There is no single amount of transliteration detail that every reader needs. The Library of Congress maintains a formal Greek romanization table, but literary translators are not obliged to make a novel-sized poem look like a library catalog.
Another route runs through Latin
Many Greek names entered English after centuries of use in Latin. That route can produce a form that is more than a letter-for-letter transliteration.
The cleanest example is Odysseus. The Greek name is Ὀδυσσεύς, conventionally rendered Odysseus. Latin writers used forms including Ulixes and Ulysses. The Perseus Dictionary of Greek and Roman Biography and Mythology records exactly that distinction.
English inherited both branches. A modern Homer translation is likely to say Odysseus. Older English literature is full of Ulysses. They are the same Ithacan.
Ulix itself takes its name from this Latin branch of Odysseus's name. The spelling history has had a reasonably long afterlife.
Herakles provides another obvious example. Greek Ἡρακλῆς appears in English as Herakles or the conventional Heracles; Latin gives us Hercules. Perseus lists the Greek Heracles and Latin Hercules together.
Ajax looks less different than it really is
The Greek name Αἴας can be transliterated Aias. The familiar English Ajax comes through the Latin spelling Aiax, conventionally printed with j in English.
This is why a translation that uses Aias can initially feel as though it has introduced a new minor warrior. It has actually removed some of the Latin and English history from the spelling.
Translators are making a readability decision too
A translator who writes Achilles, Ajax, and Hercules is using forms already deeply naturalized in English. A translator who prefers Akhilleus, Aias, and Herakles makes the Greekness of the names more visible.
Neither policy is automatically more accurate in every useful sense.
A highly transliterated name can preserve information about Greek spelling and sound while making a familiar figure look needlessly foreign to a first reader. A thoroughly Anglicized or Latinized name can be effortless to recognize while obscuring how far the English form has traveled from Greek.
The choice also affects texture. A translation in which Zeus, Aias, Akhilleus and Hektor stand together feels different on the page from one populated by Jupiter, Ajax, Achilles and Hector, even before the sentence around them changes.
Do not assume every Roman name is merely a spelling variant
There is one important limit. Greek and Roman cultures did not simply maintain two spelling systems for an identical set of gods.
Odysseus/Ulysses is a naming history for the same legendary hero. Herakles/Hercules is likewise a Greek hero received into Roman culture. But Zeus/Jupiter or Ares/Mars involves Roman identification and adaptation of deities with their own religious histories. Those pairs should not be treated as though a copyeditor changed the spelling.
Names carry reception history as well as letters.
What should a reader do?
Mostly, learn the handful of variants that occur in the edition you are reading and keep going.
If a name suddenly looks unfamiliar, check whether it is a more Greek transliteration or a Latinized form before assuming you have forgotten a character. An edition's introduction often explains its naming policy. Comparing two translations can also make the pattern obvious very quickly.
The variation is not noise around the classics. It is a small record of how the classics reached us.
A relevant Ulix tool
Breaker of Horses
Breaker of Horses includes Homeric Greek beside the Butler, Pope, and Chapman translations of the Iliad and Odyssey. Keeping the Greek visible makes it easier to see when an English name is a direct-looking transliteration and when centuries of translation history stand between the two forms.