1. Stories appeal to whole people in ways that arguments don't. Where a philosophical argument hits the brain, a well-told story goes to head and heart at once. And this is no bad thing. I have heard people say that apologetics and evangelism must be done dispassionately, lest we seem to appeal to a person's emotions; we want people to believe, not just feel. There's something in that, but I would propose that there is such a thing as emotional truth as well as rational truth, and that it is only Kant (him again!) which prevents us from seeing the former as just as important as the latter.
2. Stories involve people. Arguments, for most people, are spectator sports, but you can't help being drawn in to a well-told story. That is valuable epistemologically. The Enlightenment worldview wants us to see the individual as isolated, surrounded by data which he or she can analyse. In reality, truth is not something outside us to be discerned and analysed - we live in truth, in the same way that we live in stories.
3. Stories bring a more subtle challenge. An argument for the historicity of the resurrection based on an analysis of the evidence has value, but a story of the early church and the way the first Christians lived and died has more value. Stories are not so confrontational, and thus win a hearing. But they do nevertheless challenge!
4. Stories join together. My personal story links into many bigger stories, all of which link into the gospel story. Testimony, apologetic, evangelism - much more closely related than we have often thought, I suspect. Of course, the Enlightenment worldview within which we operate privileges 'universal' stories - and in reaction, postmodernism favours the individual story. Might it not be a powerful apologetic in itself if we can show that there is a genuine joined-up story?
5. The gospel is a story. This is the most obvious one!