Without going into the level of detail in the post, Vallicella describes the difference between eliminative materialism (which says that mental states don't exist) and identative materialism (the idea that mental states are equivalent to specific physical configurations in the brain, he did not use this exact term). After casting a jibe at the sanity of the eliminativists, he then proceeds to try to equate the two positions. Apparently he finds the identitarian position too difficult to rebut standing alone, so it must be tied to the more radical position.
On the face of it, there is a difference between saying that X does not exist and saying that X = Y. For the latter claim seems to presuppose that X does exist. But when X and Y belong to disparate categories, the difference appears to vanish. For example, is there any difference between saying that God does not exist and taking the Feuerbachian line that God is identical to a unconscious anthropomorphic projection? Suppose someone says, "God exists all right — it is just that God is an anthropomorphic projection." The proper response to such a person is to dismiss such obfuscatory rhetoric as tantamount to the claim that God does not exist. For God is not the sort of thing that could be a projection. Whether or not God exists, the concept of God is the concept of something that is a se, from itself, whence it follows that God cannot have the status of an anthropomorphic projection. So a claim that God does have this status is a claim that God does not exist.
Similarly, is there any difference between saying that mental state M is token-identical to physical state P, and saying that M does not exist? If M reduces to P, and M has all and only the properties possessed by P, then all there is is P, and the reduction is tantamount to an elimination.
The first paragraph is a poor analogy, simply because Vallicella must create an opponent who confuses whether God exists with whether the concept of God exists in order to make Vallicella's point. A putative God would be external to the mind of the opponent, and the mistake of equating it to an internal construct seems a little too elementary for the usual sphere of discussion Vallicella finds himself in.
As for the second paragraph, this is a very poor equivocation. Certainly, the identity theorist does eliminate facets of the ontological status of mental states with respect to the position of the dualist, so an elimination has occurred. However, not all eliminations are equal,and the identitarian position does specifically say our mental states are exactly that. Eliminative materialists would argue that almost all of our understanding of the mind will need to be re-worked, while identity theorists don't see the need for that sort of overhaul. It's one thing to say that happiness is the equivalent of a specific cascade of neural interactions, it's quite another to say that happiness is a false understanding of the workings of the brain.
On a personal level, I tend to accept identitarianism. I believe that when I am happy, angry, thirsty, etc., this is saying something real about me.