No spoilers included
They changed the story fairly drastically from the book. I expected that, because the story in the book is too complicated for two hours, at least for two hours that include lots of atmospheric shots of Jesse brooding at the water's edge while his dog watches.
In the books he carries his old backup revolver from Los Angeles at first, and then gradually begins to carry the department issue SIG. Of course, in the books he's also about thirty five years old, six foot tall and 175 pounds, which doesn't really describe Tom Selleck very well.
By the way, a related series of books by Robert Parker, the Sunny Randall series, also has some appearances by Jesse Stone. Randall is a Boston private eye and her character was invented with Helen Hunt in mind.