In the original comic, Ozymandius creates a fake monster and teleports it into the middle of NYC, killing a million people and making the world think it is under Alien attack. At that point the US and Russias back down from their crisis to work together to fight the threat.
In the movie version, he makes the world think that Doctor Manhatten has gone crazy and attacked cities all over the world.
The Zach version is stupider, first because since Manhatten was an agent of the US government, the rest of the world would blame America. It would probably have made war MORE likely, not less.
The rest of the movie plays out the same... the other heroes agree to go along with the deception to save the world except for Rorschach, who is vaporized by Dr. Manhatten to protect the world.
A lot of the other changes were minor and more extreme, such as fight scenes that go on for minutes that were resolved in a few panels in the comic.