Is spirituality different than superstition? Since Enlightenment we seem to have entered an age where “reality” is often confined to empirical explanations. Yet the Bible speaks of a spiritual reality that is not open to empirical investigation. 

Are there any generally accepted (within Christianity) writings on this subject?  If defining "reality" is a branch of philosophical thought, who are the writers of such thought and are there any summaries of their perspectives?
The strictly empirical worldview seems reductionistic to thoughtful Christians (not to mention many atheists etc.) because, in their view, it fails to adequately account for phenomena such as love, beauty, reason, morality, the "unreasonable effectiveness of mathematics," and the like. Moreover, empiricism as a worldview seems to fail on its own terms because it is open to explanations for observable phenomena such as a multiverse and potentially undetectable forms of matter that are themselves not open to empirical investigation.

If we thus take spiritual to be non-material or non-empirical, then superstition is merely that subset of non-material things that you think are not real. But that's being a bit coy, because Christians believe there exists, not just non-material "stuff," but also non-material personal agents -- indeed the personal agent -- who interact with the material world in various ways.

Why do we believe such things? Christian philosopher Alvin Plantinga argues that one reason we do is because of a built-in sensus divinitatis (SD), a natural faculty common to all humans that leads us to the divine. If God made us, it seems reasonable that he would instill in us a faculty to draw us to himself. Like the other faculties (sight, hearing, reason, imagination, the affections, etc.), however, the SD operates at varying levels of quality among different persons. An errant SD may lead one to become what Calvin calls a "factory of idols," seeking the spiritual in ways where it cannot be truly found -- superstition, if you will.

Moreover, it stands to reason that if God did make the world, there would be some clues in it that would point in his direction. Christians believe they can and have recognized such clues in the order, design, and telos of the world and of mankind itself. But make no mistake -- there is no neutral ground here. All facts must be interpreted, and evidence or data itself cannot act as a neutral arbiter between the interpretations (as non-Christian Stanley Fish says, "evidence is never independent and is only evidence within the precincts of a particular theory"). Thus, there is no airtight proof or evidence for God or spiritual reality that will convince all reasonable people, but a cumulative case of clues can be mounted for those with eyes to see it, as it were. (See Tim Keller, The Reason for God, chapters 8 and 9.)

Christians do not believe they have the power to "define" or delimit what is real, as your question seems to imply. We seek only to acknowledge what is real in the world where we find ourselves. As finite beings, we may not be equipped to fully explore all that exists. Hence there may be some realities we cannot explore effectively with the tools at our disposal, but we could still learn about these realities indirectly from someone who knows them directly. Hence we find ourselves seeking divine revelation to know something about non-empirical realities, and virtually all Christians believe that the person, work, and teaching of Jesus and the Bible hold such revelation to one degree or another and reveal God and his character and nature.

The three previous paragraphs, intrepid readers may note, align with John Frame's triperspectival approach laid out in his Doctrine of Knowledge of God and represent the existential, situational, and normative perspectives, respectively. Each of these is mutually interdependent, informing and being informed by the others.

Two other writers worthy of note here are particle physicist-turned-priest John Polkinhorne, who has compared and contrasted how science and theology pursue truth in their respective domains, and chemist-turned-philosopher of science Michael Polanyi, who described "tacit knowledge" (that we know more than we can tell or prove).

As a footnote: Beyond more philosophical musings like the above, which can sometimes be too abstract, I might recommend C. S. Lewis' Space Trilogy for an imaginative telling of a Christian view of reality. In it, the material world, for instance, has a maximum speed of the speed of light, whereas the spiritual world's minimum is the speed of light, and the spiritual beings interact with the material world only loosely, passing through conventional matter as we do through a fog.