When the Sub crew first arrive in Melbourne there's a building that has a hole in it and is burning inside, but later on in the movie at the train station you can see the same building in the background without a scratch on it.
The Route 3 W-class Melbourne tram that Moira and Dwight use to travel to collect the Ferrari is travelling on the wrong side of the road. The tram stop is clearly visible beside the tram as well as the tram tracks for the correct side. Also, if it's destination is Caulfield Railway Station (as on the Indicator Board), it is also travelling the wrong way as the tram is heading north up Queen St Melbourne, and Caulfield is south.
When the sub departs on its mission north, it is depicted remaining surfaced until radiation levels start rising. In real life, they would submerge just as soon as they were clear of the Australian Coast - a surfaced nuclear sub is barely seaworthy and can't go nearly as fast as it can submerged (and the story did emphasize time was of the essence). Also, submerging and surfacing are not nearly as dramatic as this movie depicts. Back in the early to middle 20th century, submarines like the German U-Boats were shaped like surface ships, and worked more efficiently on the surface, but that changed around the time the book was written - since then, submarines move most efficiently underwater.
When they walk around in protective suits in Anchorage their only concern is their limited air supply. Suits do not protect from gamma rays, they only stop getting yourself contaminated with radiation emitting particles. Thus the time being exposed to radiation should be more critical than air supply.
For various scenes, the submarine is incorrectly depicted as: 1. Flying the American Flag from the Sail (tower structure above hull) while tied up at the pier, and 2. Not flying the American Flag while underway (moving) surfaced. Ships and submarines throughout the world: 1. Fly their national flag from their Stern (back end) while tied up at the pier, and 2. Fly their national flag from their Sail (submarines, surfaced) or Mast (surface ships) while underway.
The U.S.S. Charleston is supposed to be a modified Los Angeles class with a caterpillar drive which uses magnetic coils to accelerate water down a tube resulting in a jet-like method of propulsion with no moving parts. Yet when the submarine is headed back north on it's mission an underwater stern shot shows a normal multi-blade prop.
When the submarine is first greeted at Melbourne, the Royal Australian Navy (RAN) Guards at the Gangway greet the Captain by pulling back the bolts on their assault rifles. This dangerous gesture is highly improbable.
Chief Warnzeniak is incorrectly shown wearing Gold (Officer) Dolphins. While Chiefs do get to wear khaki uniforms and a gold belt on their blue coveralls (like Officers do), they still wear Silver (Enlisted) Dolphins.
While investigating San Francisco through the periscope, the Captain incorrectly addresses Swain as "Mr. Swain". Usually only Officers that are Lieutenants or lower (like LT Garcia) are addressed as "Mr. ...." (e.g. Mr. Garcia). Enlisted men are usually addressed by their rate (e.g. Sonarman Swain, STS2 Swain, Petty Officer Swain, etc.).
While submerged, the USS Charleston is shown in some scenes to be a Los Angeles-class attack sub with VLS, while in others it appears to be an Ohio-class boomer with it's flat-surfaced missile compartment aft of the sail and unique stern plane and rudder arrangement.
Whenever we see CMDR Towers or his XO in their white uniforms, they are wearing a Unit Identifiction Mark on their right shoulders (the black and white crescent shaped item). Officers do not wear them on their uniforms. They are strictly for enlisted personnel.
When the submarine is first greeted in Melbourne, Royal Australian Navy (RAN) Admiral Cunningham salutes Commander Towers with his palm facing downwards. This is correct for the RAN (and all Commonwealth navies), which does not employ the outward facing palm salute used by the army and air force, as per naval tradition.
In the final scenes where Commander Towers walks around the hill at Point Nepean (Port Phillip heads) to join Moira Davidson as some of the last few people alive in Australia, in the background is a fully-laden container ship about to enter Melbourne's Port Phillip Bay.