Is there any difference in the rewards you would get if you surrender the quest Vs run out of moral?
Surrendering Vs Dying
According to the Community Rules Reference Guide, page 24 ( http://penultimate.de/crrg/CRRG_1_1_1.pdf ), Morale gets replenished when the heroes visit the city. Although I have not tried it myself, I think that it makes no difference, either way you loose that quest.