If you manage to capture the marshal of the opposing faction, then it'll be much harder for them to mount a serious assault on something until their king appoints a new marshal, which leads to...
If you capture their king and hold on to him, you can prevent their side from declaring peace if you want a long, drawn out war.
If you capture the marshal, another will be appointed. More damage is done to his reputation by the fact he led the army into defeat as the lords of the realm start to distrust his ability, this can eventually lead to a series of bad marshals with a series of large scale defeats, effectively crippling their armies.
Capturing the king however does NOT handicap their faction in any way other then less troops, they still hold feasts in the kings court, they still offer peace and they still assign marshals.
Just 2 days ago, I was fighting Swadia in a long drawn out war to assist my idiot allies under Graveth's banner, I captured their king and dumped him in my "kings jail" (I keep the defeated enemy kings in my own jails before I wipe out the faction).
While he was in jail he made peace with my allies, and shortly after proposed peace with me.