Throughout the entire movie (more specifically during the life boat scenes) captain Phillips' facial hair changes constantly, going from beard to goatee to clean shaved etc... The final scene when he is rescued, he has very little stubble and clean trimmed side burns when he was just on a life boat for 3 days.
At the beginning of the medic scene, the medic cuts off the yellow Navy shirt Phillips had been wearing and puts it on the counter. Later when they pan down his body, you can see the yellow shirt sitting around his waist, as if it had just been cut off his shoulders, not cut completely off.
Just after the Navy SEAL leader identifies himself as the new negotiator, the pirate has been holding his pistol to Phillips' forehead so hard that it has left quite a mark. Seconds later when Phillips is shoved back into his seat, the mark is not there.
Captain Phillips' accent seems to diminish and come and go periodically as the movie goes on. By the end of the movie he has no accent when in the beginning you could hardly understand his heavy New England speech.
At the chase scene, the navy declares that the boat is in the direction of 035. If the boat was heading to Somalia's beach, 035 would be just the opposite way back home as it heads north-east, and they were west bound.
At the beginning of the film, Capt. Phillips and his wife are portrayed driving from their home in Underhill, Vermont to Burlington International Airport (BTV) via a four-lane highway and passing an exit numbered 29.There are no four-lane highways in Vermont. It is unlikely they would have traveled on any highway to get to BTV. The only highway in that part of the state is Interstate 89, which does not have an exit numbered 29.
Life boats are equipped with food and water for a week or more for everyone on board. Yet after only 2 days, the 4 passengers are completely out of supplies and the brown supply box appears to have never been opened in the background.
The vehicle that drives Capt. Phillips to the Salalah port in Oman is shown as right side steering. Oman drives on the right and most vehicles have left-hand drive. However, the Middle East has extremely well developed Japanese second-hand imports market. Some of those cars go to Africa but quite a few remain in the region.
In a nighttime shot, you can clearly see "Alexander Maersk" painted on the side of the vessel. Nine weeks were spent filming aboard the Alexander Maersk, a container ship identical to the Maersk Alabama.
When the pirate boat is trying to enter the Maersk the captain orders a few hard maneuvers to the left and right. We get to see a shot of the rear rudders turning. When he orders to go right, you see the ship turn right on the screen from a top-down perspective. A ship that uses his rear rudders to go right will sway left, thus coming closer to the pirate ship rather than moving away from it.
The same mistake is made when he orders to turn left.
The goof items below may give away important plot points.
After killing the pirates, the SEALs pick up their gear and put it in their bags. Later on, after Captain Phillips is cured in the infirmary, an outside shot of the ship shows the same snipers picking up their gear again.
When the Navy SEALs take their synchronized kill shots on the pirates in the lifeboat, the sound is clearly that of a non-suppressed rifle. Afterwards, a suppressed rifle can clearly be seen as one of the SEALs stands up.
At the denouement just before the Navy kills the pirates, the negotiator orders to 'stop the tow.' The last pirate to come into view of their weapons is standing at the front of the lifeboat. The lifeboat is also being towed from the front. When the navy stops the tow, the pirate is shown accelerating toward the back of the boat, into view of the windows. This is inaccurate. By stopping the tow, the pirate should have been thrust forward, not backward.