Well, being a building size monster, it's only natural for there to be collateral damage. In fact, I think Godzilla caused less damage than superman, not to mention that at the point the city had already been evacuated, with the exception of the few trapped under the rubble, in which he ensured their safety, while in Man of Steel you can clearly see people standing in the open in the city while Superman and Zod were fighting and wreaking buildings. Superman should have known better than to bring the fight deeper into the city where people still haven't evacuated. He did the same thing during the fight in that small town. He hits a foe into a gas station at the speed of a bullet, before anyone has any time to know what hit them.
The thing about Godzilla is that you expect him to cause damage, but in that movie he was very restraint. He didn't purposefully knocking down building just for the sake of it, and he wasn't after the people, but the MUTOs. Despite basically being a giant animal, he had a level of intelligence, character, and dignity that you never expected from a giant monster. Because of this, you're actively rooting for him.
With Superman, however, you expect him to do the right thing and try his best not to endanger the lives around him. He could have taken the fight to a less populated area away from the city, but didn't. He should show concern about the people's lives and the property damage, but never does. You expect him to contain the damage, not add to it. This makes him look irresponsible, and not particularly someone to root for.
Godzilla succeeded at being Godzilla and then some, but Superman failed at being Superman.