When understood in this way, we immediately realize it's possible to be very religious, but not spiritual (my father is a good example of this), or quite spiritual, but not at all religious (my mother-in-law is like this); and of course, it's possible to combine characteristics of both.
We also see that spirituality has much to do with one's capacity to be present in life--that is, to pay close, non-judgmental attention to what's actually happening via one's immediate experience of the body, on a regular basis (as most of you may know already, this kind of awareness is called mindfulness).
Finally, being mindful helps us realize that we are not the same thing as our thoughts, nor the feelings that attend them. Indeed, spiritual people recognize that they can actually be the observer of their thoughts and feelings (suggesting that their "selves," whatever that is, comes from an even deeper level of conscious awareness). This increases their equanimity or even-mindedness, allowing them to avoid being held hostage to whatever thoughts come careening into consciousness. In short, spirituality doesn't necessarily have anything to do with any religious doctrine or dogma; rather, it has to do with one's state of being.