I can't say this is backed by anything, i don't know, but my gut feel is humans are somewhat different in that they have developed language both spoken and written to such an extent, that now there is a constant journaller as part of most people's minds documented everything we do, feel, think. This journaller basically happens autonomously, without direction. This journalling mechanism, i think, is what most people consider consciousness.
I suspect we fool ourselves into thinking this journaller has any control over our situation and so we give it more value than it has. I don't see how it would give us free will, for instance. But it is a powerful illusion, none the less.
I've had lots of dogs, horses, cats, and think they all are on the consciousness spectrum, but i'm guessing (couldn't possibly know) that they lack this constant overarching journaller in their brains. For species that have a richer communication, i still don't think they would have this... I think the need to write down language, and make it available to others asynchronously plays a part in it.
Interestingly, some people report that they have no such journaller, or sometimes called inner voice that talks to them. I have no way to know what it's like to be them, or if my understanding of what they are saying is indeed what they are saying, but if true, i wonder if these folks are more like what other animals feel in their heads. I have no clue. Frankly, i can't imagine that someone wouldn't have a journaller in their heads, and so it's even hard for me to imagine. I assume i just don't understand.