Pew surveys of the society and its changes use the term "spiritual but not religious" to mean that the person considers him or herself to be not affiliated in an organized, established body of believers and yet does remain receptive and maybe responsive to something bigger than human powers; something worthy of respect and seeking after (worth-ship or worship); and accepts some point of reference to remind her or himself of humbleness and human dignity.
There are clergy, ordained or lay, who also make a distinction between Christianity and Church-ianity (or Church-inanity). That is to say, overzealous of rituals can redirect one's attention and love from the Spirit of the Law to the Letter of the Law, similar to the imagery in the Bible to describe "white-washed tombs" that are righteously bright and bold on the surface, but which contain only death on the inside. In other words, organized religions of any civilization can blur the lines between the form but lose sight of the intention and heart of the teachings and the way of living that the particular body of belief and believers aim to embody day in and day out.
The illustration attached here comes from translate.google.com when typing 'spirit' on the English side and calling up the Hebrew words that cover each sense of the word. Clearly, there are many different words in Hebrew, and thus many distinctions or facets of meaning to mark with differing word roots and pools of meaning. This example of the many sides of 'spirit' suggests something about the "spiritual but not religious" phenomenon. Even before that expression came to be, perhaps there have always been people unwilling to commit to the forms of organized religion available in their society. Observers have said that it is the most skeptical people who take longest to come to a decision about their relationship to religious practices that come down the generations. Very often these are the people who turn out to be the strongest advocates for that religion if/when they do come around to it. By contrast, those who are born into the body of belief may take everything in like Mother's Milk, without giving the matter a lot of thought and who are satisfied with the sense of belonging and habit that comes from unexamined acceptance; without scrutiny.
Before the phrase "spiritual but not religious" was coined, those people would keep one ear open to the institutional religion and the other ear open to other sources of guidance, truth, and respect. They could see value and meaning and truth in much of what they heard in both of their ears, but would not grant exclusive truth value to one source or to the other. This makes sense from a logical or rational way of seeing things: how could Ultimate Truth be contained or comprehended by one exclusive set of words and ideas, one religion over another? Yet in order to speak the language of spiritual things, a fixed set of vocabulary and ideas has to be engaged with: it matters less the particular religious language that you adopt and gain fluency in than the fact that you do engage with some specific body of belief and not remain idle in limbo, paralyzed from making a commitment by the abundance of choices and possible truths.
As the long set of Hebrew nouns for 'spirit' suggests, there are many aspects of the spiritual world and spiritual maturation and development that connect with this topic. Being open, hearkening, and seeking after truths in the span of a single lifetime is a fundamental part of building a life of meaning and humanity. For many people there is advantage in clinging to an elaborated and established form of religious tradition or community of faithful people. But for many others there is self-criticism (I am not worthy enough or righteous enough), lack of trust (unable to rest in the truths), anxiety about committing to a single path and foreclosing other ways, or a distaste for the sense of exclusivism that draws boundaries between self and others. And so the category of self-identifying as "spiritual but not religious" continues to grow, while the organized, institutional religious shrink to a smaller but more committed core of purposeful followers of their own faith community's ways. The delicate balance remains a living thing: between grasping confidently and firmly one exclusive way to worship and grow, on the one hand, and yet loosen that grip to allow other things to come to hand, on the other hand. Put another way, it is important to speak confidently and in full trust in what is righteous, but at the same time to remain humble and openhearted, ready to listen and to care. In other words to go forward, one has to know the truths of all ages, and also not know things with any finality; to be certain but also uncertain. Definite but not foreclosing other views.