In 1861, the US issued an announcement that they were looking for designers to submit plans for an ironclad ship. They did this in response to information they were obtaining indicating that the South had already begun building their own ironclad ships. Realizing that there was no way they could defeat an ironclad navy with their own wooden ships, the US Navy decided to pursue their own. The winning design was one that was put together by a man named John Ericsson. His design had actually been submitted originally to Napoleon III in 1854, but Napoleon decided against it. After Ericsson convinced the US Navy that his ship was sound and could be built quickly, they decided to go with his design. This resulted in the construction of the USS Monitor. The monitor was designed as a ship with two guns placed side-by-side, hence the revolving turret, which was basically the only portion of the ship that was visible above water. This also presented a smaller target for opposing ships, making it more difficult to hit.
The CSS Virginia was built from remnants of the USS Merrimack, which had been salvaged by the Confederates in early 1862. The intention with this ship was to model it on current ship designs, with the bulk of the ship remaining below water. The topside would be covered with thick layers of iron to protect the ten guns that would be deployed. The Virginia was designed with the intention of becoming a wooden ship destroyer, and the Southerners believed that by having ten guns they would be able to dispatch the wooden ships of the US Navy pretty easily.
Ultimately, the primary reason that the USS Monitor was so different was simply that it was designed by someone who did not have a traditional naval background. He decided to think outside the box and came up with something less traditional. It is a credit to the US Navy that they were willing to take a chance on something so radically different.