Non-dualistic spirituality posits that our true identity is not with our individual minds and bodies, but with an underlying Self, the serene space in which all these minds and bodies appear and disappear and their underlying substance.
What is the difference between that Self and God? In traditional Hindu philosophy, the Self as it is is called "Brahman." In Buddhism it is called sunya, or the void (though many Buddhists would disagree that sunya should be called the Self; that's an argument for another day).
The Self is beyond creation, destruction, and all opposites. All that may be said of it is that it cannot be described.
Now, as it manifests in the world (how does this happen? it remains the greatest mystery), it may be considered the creator and the destroyer, etc. In that capacity it is called God.
But strictly speaking the Self is beyond even God. And when we realize our true identities -- it is with the Self, and not with God, except inasmuch as God is also the Self.
Enlightenment does not mean feeling like you have the total control over the universe that God does. It means the discovery that the universe itself is only an appearance in Something Else where issues of such control do not arise. The discovery of that Something Else is what we are aiming for.