Eliminativism (in particular, Eliminative materialsm) proposes that the theory of 'mental states' ('folk psychology', hereafter FP) is a theory. This is part of a wider position known as Theory-theory, as @Era points out. Moreover, it states that this theory is false and should be completely replaced (by neuroscience or something similar). Churchland describes the belief in mental states as follows:
The problematic conviction that another individual is the subject of
certain mental states is not inferred deductively from his behavior,
nor is it inferred by inductive analogy from the perilously
isolated instance of one's own case. Rather, that conviction is a
singular explanatory hypothesis of a perfectly straightforward
kind. Its function, in conjunction with the background laws of folk
psychology, is to provide explanations/predictions/understanding
of the individual's continuing behavior, and it is credible to the
degree that it is successful in this regard over competing
Your point about the usefulness of 'higher level' descriptions is one addresed by Paul Churchland very early on in the history of Eliminative Materialism, and constitutes a position known as functionalism, the origins of which probably lie with Wilfrid Sellars:
Perfect theories, perhaps, have no need to evolve. But FP is
profoundly imperfect. Its failure to develop its resources and extend
its range of succ ess is therefore darkly curious, and one must
query the integrity of its basic categories. To use Imre Lakatos'
terms, FP is a stagnant or degenerating research program, and has been
In other words, Churchland would say you are wrong in claiming that mental states are 'useful'. He points out that there are many limitations and flaws to the theory, that from a scientific perspective render it not very useful.
Churchland calls this approach 'functionalism', and compares it to those who argued that Alchemy was a useful abstraction and not an ontological theory, and so it shouldn't be replaced by chemistry.
In summary, when confronted with the explanatory impotence, stagnant
history, and systematic isolation of the intentional idioms of FP, it
is not an adequate or responsive defense to insist that those idioms
are abstract, functional, and irreducible in character. For one
thing, this same defense could have been mounted with comparable
plausibility no matter what haywire network of internal states our
folklore had ascribed to us.
The quotes are taken from this paper:
Elimanitve Materialism and the Propositional Attitudes
By Paul Churchland. I recommend reading that for a more satisfying answer to your question.
To answer the question directly, you ask:
Type identity physicalism says mental states are identical with brain
states. Eliminativism on the other hand states that mental states
don't exist at all.
Isn't this just a word game? Does it really matter whether we say
"Super-Man is identical to Clark Kent" or whether we say "Super-Man
doesn't really exist, only Clark Kent does"?
You describe with dazzling precision the exact difference between the two positions. And so you ask a further question (I take it to be the real question): Isn't this a word game?. I suppose that it is, in the same way that the statements "the Earth is round" and "the Earth is flat" differ only as part of a mere 'word-game', if that refers simply to the fact that 'round' and 'flat' are words.