Achilles wasn't scared to die. He even makes the choice to die in war. Achilles' mother tells him that he has a choice: he may not fight in the Trojan war, marry and have children, and be loved by his family, but his name will be forgotten within a few generations of his death.
Or, he may fight and be killed, but his name will be remembered for eternity.
As it turns out, she was correct: Achilles is still the subject of many a school lesson, and Iliad was made into a movie as recently as 2004!
Achilles chooses not to fight on several occasions, but this is not due to cowardice: he simply does not feel well-respected by Agamemnon and Menelaus.