Although the term narrative usually does mean precisely the same as the everyday word story, when used technically, the word story is often reserved for the simple chronological sequence of events that occur within the narrative. The word narrative is related to the word 'narrate' - a narrative is a story told in a particular way
What makes narratives so engaging is hard to pin down but it is certainly linked to aspects of their structure and form.
The form of a typical narrative involves an individual (called the protagonist - a character who is presented as some kind of 'hero' and on the side of right); this protagonist is brought to face a conflict that disturbs the equilibrium of life. The conflict is created by another character (or a system) called the antagonist presented as being on the side of wrong, i.e. as a 'villain'. There is usually, too, some form of 'helper' object that aids the protagonist.
In a typical narrative, events are related in a sequence using a structure that seems convincingly to move along a course of cause and effect; these linked events form a natural seeming 'beginning>middle>end' to the story and move quite naturally into such aspects as the narrative climax, itself followed by a satisfying resolution.
We feel engaged and compelled by this form and structure; we easily relate to the hero figure; we become fascinated by the tension created by wondering 'what will happen next'; and we want to know 'where it will all lead'.