Organized religion is quite different, and the explanation is right there in those two words: organized. Organizations can do so much more than individuals can. Up until about 200 or so years ago, there was no difference between church and state. Religious authority was secular authority. The church was the town hall—the meeting place where all the business of the community was conducted.
As such, religions did everything that we, in the US, now think of as a secular activity. They ran schools. They brought the community together. They helped form cooperative efforts such as home-building and farming and whatnot. The industrial revolution sounded the death knell for a unified religious and secular authority. The separation had been in the works for a long time, but once people were moved into cities to work in factories, uprooted from their home communities and thrown together with all kinds of strangers who spoke and thought differently, there could be no unity that is required for religious experience.
Religion has so many rules, just like secular laws. It helped gain cooperation with these laws if religious/secular officials appealed to the highest authority as the source of these laws. That was the wellspring for the dogmatic aspect of religion. The spiritual side reinforced this, to some degree, but for most people, spirituality was an esoteric thing, that was the domain of mystics, for the most part.
These days, spirituality is still the domain of mystics, or, at least, the mystical side of human beings. It is a numinous connection with the all—a sense of unity with all people and things in the universe. More people have access to it, these days. And religion is no longer the law of the land. It is merely an adjunct law that has something to do with community, and something to do with moral justification for law, and something to do with psychology and something to do with understanding the mysteries of life (or death).
These things are important concerns of humans, but it is more difficult to see how they affect daily life, because secular law, in most cases, mimics religious law. In some Islamic nations, they are still the same thing.
In order to gain a mystical oneness with a higher authority, one needs to practice something diligently. This practice allows a person to go past their overt thoughts into a space where their non-thinking thoughts lie. These experiences are the spiritual experiences, and they happen in so many ways, and most of them aren’t involved with organized religion. Thus spirituality has gradually become separate from both the secular and religious domains. It has always been a personal experience, but now it is even more so. A personal, mystical experience that makes people feel special.
Sometimes spirituality is connected with organized activity. Dance and music and prayer and meditation are examples of activities that gain one access to spirituality. Church rituals are designed to throw people into these kinds of states—overwhelming music, beautiful architecture, prayerful postures, ecstatic dancing, snake handling and droning voices are all part of the effort to change consciousness of a congregation. People might think they are falling asleep, but that is not necessarily the case. Some are moving into that non-thinking state of awareness.
However, most religions want congregants to pay attention! This keeps people in their heads instead of allowing an altered consciousness. So religious organizations often let their legal concerns counteract their spiritual activities. Thus, people who seek a personal sense of spirituality must do it on their own, and it begins to seem like it has only to do with them, individually, and not via their organizational participation. So, organized religion has a problematic relationship with spirituality. So, too, does it seem like spiritual reality is different from organized religion.