The greatest difference between these two characters lies in the respective objects of their devotion. Antigone is devoted to a higher cause: the cause of cosmic justice. She seeks to bury her brother Polynices not just out of sisterly love, but in accordance with unwavering standards of justice decreed by the gods for all eternity.
Achilles, on the other hand, is out for himself. All that really matters to him is his own exaggerated sense of wounded pride. While his Achaean comrades are busy being slaughtered outside the walls of Troy, he's sulking in his tent, brooding over the appropriation of his sex slave by Agamemnon. Even when Achilles finally returns to the field of battle, it's only for personal reasons, namely his desire to avenge the death of his bosom buddy Patroclus, which he takes as a personal insult.
For warriors like Achilles, glory is everything, and glory is primarily achieved in war. Achilles knows that if he remains at Troy, he will eventually be killed, but the lust for glory drives him on regardless. As a woman in Ancient Greek society, Antigone is placed in a different situation. Glory is not really a relevant concept for her, nor is she aware of the precise fate that lies in store for her. However, it doesn't take a genius to work out what will happen to her if she insists on defying Creon. Antigone knows that she will pay for her defiance with her life, but she persists in any case, fervently believing that she's honoring the gods.
The key difference between the two characters is gender. In ancient Greek, gender roles were quite distinct, and behavior considered admirable or honorable in a man would be considered inappropriate in a woman. Thus scholars often note that it is Ismene rather than Antigone who was considered closer to the ideal of Greek womanhood. It also should be noted that Achilles is an example of a Homeric hero, with Mycenaean origins seen through the lens of the archaic period (ca. 800 BC), while Antigone dates to the classical period, which had a different understanding of the role of complex philosophical and theological understanding in guiding behavior.
Both Antigone and Achilles are similar in letting their individual judgments guide them to unpopular stances. Achilles, however, acts out of wounded pride and a sense of his own honor while Antigone is motivated by religious duty. Achilles desecrates the corpse of Hector while Antigone is morally and religiously convinced that leaving a corpse unburied offends the gods. Achilles is concerned with his public reputation or fame (Greek: kleos) while Antigone's defiance is secret and concerned primarily with obedience to the gods and family duties. Achilles acts from a position of power and Antigone from a marginalized position as a woman and a daughter of an incestuous marriage.
Achilles, as portrayed in Homer's Iliad, and Antigone, as portrayed in Sophocles' Antigone, face some similar challenges. One issue both face is having to deal with a king who has authority over them, but whose authority they do not respect (Agamemnon in Achilles' case; Creon in Antigone's case).
Achilles and Antigone also have to deal with the death of a loved one. Patroclus is Achilles' best friend, whereas Polyneices is Antigone's brother.
I'm not sure that code of honour, pursuit of glory, and fate are really comparable in the situations of Antigone and Achilles, though.
For one thing, honour for men and women in ancient Greece was gained in two different realms. For men, honour was gained in the public sphere (good counsel in public and good fighting on the battlefield), whereas a woman's honor was acquired in the home (especially maintaining faithfulness to one's husband). Antigone violates what was expected of women by going outside the home to bury Polyneices. Antigone's presence in the public sphere is one of the things about her behavior that irritates Creon. In some sense, though, I suppose we could say that Antigone becomes like Achilles by going into the public sphere and burying her brother, who had fallen on the battlefield.
Regarding the pursuit of glory, while it is clear that Achilles' return to battle is motivated by the pursuit of glory, Antigone is motivated by divine law as opposed to human law ("Zeus did not announce those laws to me"; line 450 in Ian Johnston's translation). Antigone does acknowledge that she can gain glory by burying her brother (line 502), but I'm not sure glory is her primary motivation.
As for fate, Achilles is quite aware of the fate that awaits him if he remains at Troy (death, but glory) or if he leaves (old age, but no glory). Antigone, however, does not appear concerned with fate. Unlike her father Oedipus, Antigone does not seem to have any prophecy hanging over her head regarding her fate.