Homer's character development was profound, so there's much to be said about these two characters. Achilles and Hektor are similar in some ways, but very different in others. Both were the greatest warriors of their respective armies — Achilles for the Achaians (Greeks), and Hektor for the Trojans.
As with most Greek epics, both men also exhibit human flaws. Hektor, despite being the strongest of Troy's warriors, could be impetuous at times and cowardly at others. Hektor's pride was also a flaw (one that eventually led to his death). Achilles was angry and stubborn from the beginning of the story, when he refused to fight because Agamemnon dishonored him. Rage and vengeance carried Achilles through to the end of the epic, as evidenced by his mutilation of Hektor's body.
Despite these similarities, Achilles and Hektor were different in many ways. Hektor was purely mortal, the son of a king and queen. Achilles had some divinity in his blood (his mother, the nymph Thetis, was the daughter of a sea god). When Achilles was a boy, Thetis tried to boost his immortality by feeding him ambrosia (the food of the gods) and dipping him in the river Styx. In battle, Achilles had aid in the form of divine intervention: help from the goddess Athena.
Hektor was a loving family man (a husband, father, son, and brother). His family always remained very important to him. Achilles was an only child who never wed nor fathered children. He sought one thing: victory and praise on the battlefield.
Hektor was a team player; a gifted commander as well as soldier. His sense of duty and leadership made him refuse the wine that his mother brought him, out of fear that he was too tired and it would make him forget his duty to his troops. Achilles, on the other hand, was a loner, and an unreliable leader who either pouted or turned violent when he didn't get his way.