An epic hero is supposed to be greater than all other men, descended from the gods. If Beowulf is so, than he must prove that he is strong enough to take on a supernatural being on his own strength. To use a weapon would be to admit that he needs help - and admitting that is a struggle for him throughout the story. This is why, in Beowulf's last battle, he must finally accept the help of Wiglaf. This shows that he has grown as a person, exchanging his own hubris for the good of his people. This underscores Beowulf's wisdom, which works to make him more of a hero.
Of course, Beowulf has also basically been dared by Unferth to prove his own abilities:
I anticipate worse luck for your adventure—though you've braved the blows of battle in grim struggle—if you wait through the night of Grendel's approach!”
Beowulf stands up to this comment be retelling his stories of previous conquests - and so he must follow through with what he says he did before, in order to be believed and respected:
“I say in truth that I have proved more might in the sea than any other man, and more endurance in the ocean. ...A fierce creature held me firm and pulled me to the bottom with the strongest grip. Nevertheless, it was granted to me that I pierce the monster with my sword point; by my hand and battle-blade was the giant sea-beast conquered."