Actually, both names were changed. Achilles' name in the Iliad, and elsewhere in Greek literature, was Achilleus (Ἀχιλλεύς).

However, Odysseus' name changed so much because it was influenced by the Etruscan Uluxe or perhaps the Siculian Οὐλίξης, both of which are versions/transformations of Odysseus.

Edit: I looked into it more and it seems that there is some dispute about where exactly the Romans received the name Ulixes. Some make the argument for Etruscans, others Sicilians, and still others Messapians. The point being that they received the name second hand.

Another interesting point is that the first Roman to equate Ulixes with Ὀδυσσεύς was Livius Andronicus, who is one of the earliest Roman poets that we know of, although we now only have fragments. Livius wrote a translation of Homer's Odyssey in Latin and used the name Ulixes to "translate" the Greek Ὀδυσσεύς. What's significant about this is that Livius, unlike the later Ennius, was often not happy just to transliterate Greek names, but instead chose Latin or native Italic names to serve as equivalents. The best example of this is his use of the local goddess Camena instead of the transliteration Musa in Homer's initial invocation to the Muse. So it would make sense for Livius to choose the native name Ulixes rather than a simple transliteration. His use seems to have stuck.

It's also worth noting that even the Greeks themselves had different versions of Odysseus' name. Here are a few: Oluseus, Olusseus, Oluteus, Olutteus, Oliseus. Different dialects of Greek formed his name according to their own phonological tendencies.

Sources: Roberto Price Homeric Whispers, Irad Malkin The Returns of Odysseus, Stephen Hinds Allusion and Intertext, LSJ