madaboutasoiaf:

Catelyn wanted to run to him, to kiss his sweet brow, to wrap him in her arms and hold him so tightly that he would never come to harm… but here in front of his lords, she dared not. He was playing a man’s part now, and she would not take that away from him. So she held herself at the far end of the basalt slab they were using for a table. The direwolf got to his feet and padded across the room to where she stood. It seemed bigger than a wolf ought to be. “You’ve grown a beard,” she said to Robb, while Grey Wind sniffed her hand.
He rubbed his stubbled jaw, suddenly awkward. “Yes.” His chin hairs were redder than the ones on his head.

Robb is “Playing a man’s part” near the end of AGOT and so he grows a beard, better to look the part.

He seemed taller than when she’d left, and the wisps of beard did make him look older. “Edmure was sixteen when he grew his first whiskers.”
“I will be sixteen soon enough,” Robb said.

By ASOS it’s not a part, not an act.

Robb stood on the dais. He is a boy no longer, she realized with a pang. He is sixteen now, a man grown. Just look at him. War had melted all the softness from his face and left him hard and lean. He had shaved his beard away, but his auburn hair fell uncut to his shoulders. The recent rains had rusted his mail and left brown stains on the white of his cloak and surcoat. Or perhaps the stains were blood. On his head was the sword crown they had fashioned him of bronze and iron. He bears it more comfortably now He bears it like a king.

He’s shaved the beard because he doesn’t need to look the part anymore. He grew up so fast, but he’s still only sixteen even if Westeros considers him a man grown. Too young for his burden, and gone too soon.