How is a play script different from narrative writing

They are much the same. Script is the more general term, and it can be used to refer to plays and to screenplays or to any written material meant for any kind of oratory or dramatic work. Here the word script will refer to screenplays. A script is the spoken portion of a project for television, film, or other kind of recorded medium. A script contains a lot of the same kinds of material you would find in a play, like general movement/blocking, suggestions of emotional content, entrances/exits, or even technical kinds of directions related to use of cameras [Reveal, for example: a character or other object moves across the screen to show something of importance behind]. Some differences with scripts [screenplays] are that the action can be filmed at widely different locales, and over the course of weeks or months [even years, as was the case for the LOTR trilogy, filmed simultaneously over roughly a 2 year period] the action can be filmed completely out of sequence for practical ease and edited later, and usually the intention is that the final edited version is the fixed and permanent version of the project. 

Plays are written and designed to be acted out in one physical location, with changes of scenery as appropriate. Live performers never actually perform the play exactly the same twice, and this live aspect adds palpable energy to stage performances.