Cloverfield worked mighty hard to hide the identity of the film's giant monster until the climax. Some were disappointed by the bizarre look of the creature, but there's no question it was one of the most intense monsters ever to take a swipe at New York City's skyline.
The reason the monster is so effective has to due with the intention the filmmakers had for the beast. They have frequently discussed how the monster is not your typical rampaging beast - instead, it is like an infant, afraid of the world it has been thrown into and the small creatures (humans) attacking it. It tried desperately to escape, but couldn't maneuver the maze of streets and buildings in Manhattan.
The monster definitely redesigned the look of the city, but it was mostly due to collisions and stumbling into skyscrapers. The only real time it seemed to knowingly attack was when it cut off the head of the Statue of Liberty.