Continuity mistake: When Captain Phillips is telling members of the Navy that he is in Seat 15, a Naval CPO writes the words "Seat 15" below "Captain Phillips" on the board. A scene or two later, the words "Seat 15" are to the right of "CAPT PHILLIPS." Also, "Captain Phillips" is now "CAPT PHILLIPS." (01:28:05 - 01:49:45)
Continuity mistake: In the lifeboat on the way to the mainland, Capt Phillips sometimes is clean-shaven and sometimes has stubble.
Continuity mistake: At the beginning of the movie, the scene where Captain Phillips and wife are in a car driving to the airport, Catherine Keener's hair changes back and forth between shots. The left side of her hair is swept back in a clip or comb. In some shots all the hair on her left is swept back, and in other shots there are loose curls hanging down. Flips several times.
Answer: If they pay the pirates, it encourages more to try the same thing. Shooting them would discourage anyone else from attempting to use a hostage to extort money.