I don't think the OP's version of 'spirituality' is typical of what people mean when they talk about spirituality. It doesn't just mean 'something greater than yourself', although that's the phrase that's thrown around. It usually means 'something *magical* greater than yourself'. It doesn't just mean 'the universe', it means 'the universe has a mind and causes things to happen for a reason'.

I've talked to people and explained to them that the universe is 'greater than myself', but they come back with 'yeah, but there's a purpose behind it all'. This is not much different from typical theistic beliefs. I think the key difference is that spirituality is more about experiences than about any particular creed. They emphasize the experience of 'feeling' that there is purpose and meaning behind what to you and me would call coincidences. A common saying of spiritualists is that 'there are no coincidences' or 'everything happens for a reason'.

I personally think there is a naturalist (i.e. non-magical) explanation for spiritual experiences. It mostly has to do with an intuitive mind-set that you allow yourself to fall back into. It is very similar to the intuitive mindset of children. Rational thinking, to a great extent, is characterized by an effort to deny the intuitive mindset, for the very good reason that many intuitions are systematically flawed. We call such flawed intuitions 'fallacies'. But when you allow yourself to be in the intuitive mindset, there IS a definite difference in the character of the experience. It is characterized by a sense of oneness, awe, wonder, and contentment. Everybody has had these experiences. Spiritualists specifically try to cultivate them. Unfortunately, they tend to eschew the benefits of rational thinking along the way. That's why the 'higher power' they emphasize is a magical mind-like one. It plays into the intuitive fallacy of anthropomorphization.