It's not like that at all. Many people consider there to be a clear and distinct separation between spirituality and religion, particularly now that organized religion has evolved into its current state. The difference comes down to seeing religion as the belief in a specific God, where as spirituality might include some belief in a larger or unexplained existence beyond what mere science can explain, or really any greater moral or philosophical identification. In some cases this is seen as nothing more than beveling there is a God, but not having anything to do with what is commonly seen as the less than pleasing parts of organized religion.
However you personally define it though, it's not uncommon, as Takeru pointed out, for people (myself included) to identify themselves as spiritual but not religious.