As a neo-Vedantist mystic who follows Sri Aurobindo and the Mother, for me, the phrase "spiritual but not religious" means that I constantly engage in an inner battle to conquer my own lower nature and allow the Divine within me to manifest itself through me. In short, the purpose of spirituality is to transform human nature so that it is more and more able to express a higher Divine nature, liberated from desire, biological conditioning and the limitations of the human mind.
This transformation is brought about through a growing inner relationship with the Divine, which transcends both the sense-perceptions and the rational intellect, for which, indeed, the senses and intellect are mere instruments. This is really the key difference between spirituality and organized religion. The former is transformative and independent of any specific mental creeds or beliefs (indeed, once you understand that spiritual experiences are transrational, you realize that you can express the existence of God using both mental affirmations and mental negations).
Religious texts, creeds and beliefs may, at their best, be an asset facilitating this transformation (or an obstacle if they are irrational and unjust), but ultimately the inner transformation and liberation that seekers strive for transcends any mental formulas that might have facilitated it.
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