The notion that one can be spiritual but not religious is like saying you have blood in your body, but don't care much for the skeleton. Similarly, claiming you need to go to Church to be spiritual is like saying you need to shop at Macy's to wear clothes. Both ideas are bunk.
I know what is meant by the people who make such claims, but I'll get to that in a bit. First. let's take a look at the terms. It matters because words matter, and understanding what words one uses is simply helpful in understanding what you mean, and deriving meaning from your understanding.
In the first place, we are all spiritual.
What does it mean to be spiritual? It means to have or be a spirit. We don't have to get deep into theology and philosophy, or make a pile of differentiated claims in order to grasp this. Often it is used to make a distinction between human consciousness and other animals. You can use that if you like. I think it's better not to make assumptions about animals or rocks (after all, it is the much-lauded St. Paul in the New Testament who describes material creation itself as "groaning" as it waits for the culmination of redemption.)
It may be simpler but right to say that spirit is what you are. It is the basis of your consciousness, including your awareness of yourself. It is you. So whatever you think, do, understand, feel, or create is associated with you as spirit, and therefore has spiritual ramifications. Communication, reading, walking, watching television, giving your child a hug, as well as going to Church and singing, praying and talking to others, are all spiritual acts.
And by the way, it doesn't matter if you think spirit stems from a non-material source, a soul, or is the product of millions of neurons firing throughout your nervous system and being interpreted in the brain, the basic definition remains the same.
Next, we are also all religious.
What does it mean to say one is "religious"? There is some controversy about where the original words come from, but that doesn't really change the way the word has always been defined, or the way it is used in the texts of varying traditions. The most-accepted definition of the word "religio" is "to bind".
Being religious means that we don't behave in chaotic ways. We do, of course, to some degree, but if one takes it too far, instability can cause disruption and distress for oneself and others, leading to mental illness. We tend to behave in patterns, develop habits, instill order and meaning into our lives on the simplest level.
All of us have rituals, whether we regularly eat a microwaved meal over the kitchen sink or regularly eat elaborately prepared meals as a family at a dinner table. Both are religious expressions in the sense that they tie or bind us to certain habits as well as to each other. Our ties can be loose or rigid.
The kinds of habits we borrow from the environment, from genes, from beliefs and from others has an effect on how we relate to other people. How we relate to other people helps to define each one of us as a distinct identity in relationship with others. As such, it reveals what we care about, what we love, what we dislike, as well as that stuff that comprises character.
You can be just as religious as you read the Sunday papers and watch the news shows and afternoon football games as someone who dresses up, goes to church, sings in the choir, prays and reads the Bible and has lunch with other church members.
Spirituality and religion are not merely married; these aspects of the human person are more like conjoined twins. You can't be one without being the other. Your religion is in many ways merely the action of your spirit from moment to moment.
I get there are other connotative meanings, I just think they are sloppily used and miss the point.
Most people merely mean that they can find meaning and enjoy the presence of God and experience beauty without going to an institutional Church. And most people invested in religious traditions claim you need the appropriate form, the church, to appropriately apprehend and understand your experiences.
There is a kind of conflict between the inside of things and the outside of things. That is the real and legitimate conflict. But it is too simplistic to assume that the inside of things can be labeled as "spiritual" and the outside of things as "religious". (It is also an error to think that these are in opposition, or a polarity, but that is for another time.)
Anyone who claims to be spiritual but not religious is still religious -- they still have a set of beliefs, have habits and rituals, and relate to others in distinct ways that inform their identity. Often, however, the problem is solipsism, not lack of religion. They conflate "spirituality" with being on the inside of things and in pursuit of spirituality risk being cut off from any reality outside their experience.
How is a religion that emphasizes the interior going to respond to the words of St. James, who says that "true and undefiled religion" surfaces in specific actions, including caring for the oppressed, and not being a self-indulgent consumer seeking the things everyone else seeks?
It seems to me that a religion that remains only on the inside of things, as if externals do not matter, can lead to narcissism. If all you care about is your own experience, or worse, your own growth (a synonym for cancer in some circles, as per the insight offered on more than one occasion by James Hillman), your religion is distorted, and you, as a spirit, may suffer. You can be so inside of things you forget there is an outside.
On the other extreme, if your spirituality is focused on externals, and you are consumed with thinking the right things, praying correctly, having all the right signs and moral codes, and are also concerned that others follow your lead, but you miss the interior meaning beneath the surface of these things, you are no better than the religious leaders who demanded Jesus be crucified. They made such demands on the basis of tradition, rigid doctrines, and the authority of those who thought of themselves as firmly aligned with their father, Abraham. Jesus agitated them further when he said their father was not Abraham, but the devil.
The point is to get to the heart of the interior/exterior conflict, and not conflate this difficulty with being spiritual or being religious. We are all spiritual and we are all religious. The way we behave as such, the way our spirituality is religiously expressed, one might say, is what finally matters.