When the rescue ship "Rudnitsky" leaves port, its name is clearly visible on the hull. But the name is in the Latin alphabet. A Russian ship would have its name in Cyrillic letters. Also, the full name of the ship "Mikhail Rudnitsky" would have been used.
The submarine weapons officer states that they are carrying SS-16 missiles. SS-16 is the NATO designation for the missiles and would obviously not be used by a Russian crew.
At around 31 minutes we see the submarine as small compared to the depth of water above it. Kursk was lying at 108 meters deep, less than its own length (154 meters).
When David Russell is in the Operations Room, one of the female lieutenants, in the blue jumper, has her rank epaulet on upside down on her left shoulder. The 'Ring' is at the top when worn correctly.
Both the equipment and the deep-sea divers who breached the hull of Kursk were Norwegian, not British, as the movie suggests.
In one scene the Royal Navy's rescue submersible LR5 is being loaded on a Norwegian ship. In the background at the dock there is a yellow truck with a flatbed trailer. The truck is a DAF XF Euro 6 model which did not exist at time the movie is set. DAF started production of this model only in 2013. The sinking of the Kursk took place in 2000.