There is a kind of epistemological ‘duality’ to our thinking about consciousness.

In 'The Puzzle of Conscious Experience', the philosopher David Chalmers describes the 'Easy Problem of Consciousness' as the question of how a cognitive agent is able to perceive things and be aware of things. This would also include awareness of self: self-awareness. This 'easy' problem seems perfectly solvable through the normal scientific methods: it is indeed conceivable that future cognitive science and neuroscience theories will provide a very good, if not complete, explanation as to how it is that such an agent perceives things in its environment, react to it, think about it, etc. Indeed, notice that this is all about functionality and causality: how, for example, the brain mediates between sensory input and motor output; how information is represented and processed, etc. And functionality is the bread and butter of science. And, reading your post, apparently James looks at consciousness from this perspective as well, when he sees consciousness as a function.

However, what David Chalmers calls the 'Hard Problem of Consciousness' is why all this functionality is accompanied by the qualitative subjective experience of consciousness. Indeed, imagine a robot driven by some algorithm and equipped with sensory devices and actuators that allows it to perceive objects in its environment, track them, react to them, plan things about them, etc. Clearly they are aware in the 'easy' sense of awareness ... but are they consciously experiencing anything? Do they have a sensation of 'red' the way we do? Or do they just react in just the 'right' ways? And if they are conscious ... why? Where does it come from? Another 'hard' question: What purpose does consciousness serve? Why do we have subjective qualitative experiences, if the 'wirings' of the brain provides us with the functionality to ensure that we can react to our environment in all the 'appropriate ways'? And note, our brains process all kinds of information unconsciously ... and we act on the results of those processes ... so what does being conscious do for us?

At this point I would like to mention the bizarre phenomenon of 'blindsight': humans who have the condition of blindsight .... are blind ... and have sight! What?! Well, they have 'sight' in all the respects of vision that we can 'easily' explain: they can react to objects and movement in their visual field in all the 'right' ways. But, at the same time they have no conscious awareness of what they see: they are 'blind'. And this really raises the 'hard' question: why are they not conscious of that particular perception? So again, why is there even consciousness, when a phenomenon like blindsight seems to demonstrate you don't need it?

It is here, I think, that we can start to address some of your questions: 'What is the difference between a conscious experience and an experience?'; Why 'add consciousness?'; 'Isn't consciousness redundant?'. Well, I think the case of blindsight, and indeed the general duality between 'easy' accounts of experience, and 'hard' accounts of experience, comes into play. For example, we can talk about blindsight people having an experience, in the sense that they 'know' that something is there. But, clearly do not have a conscious experience. Indeed, when we talk about ‘experience’, we can do this from the ‘easy’ point of view by talking about functionality, but we can also talk about the ‘conscious experience’ from the ‘hard’ point of view, which refers to the subjective, qualitative sensations that accompany that functionality.