Surely this much at least is true: The belief that P is the case (say, that my illness is gone) and the desire that P be the case are very different mental states -- the possession of one without the other explaining much human dissatisfaction. Less cleanly distinct, however, are the desire that P (or for X) and the belief that P (or having X) would be good.
I don't insist that the desires and believings-good are utterly inseparable. Maybe we sometimes believe that things are good apathetically, without desiring them; surely we sometimes desire things that we don’t believe are, all things considered, good. But I’m suspicious of the existence of utter apathy. And if believing good requires believing good all things considered, perhaps we should think genuine desiring, too, is desiring all things considered; or conversely if we allow for conflicting and competing desires that pick up on individual desirable aspects of a thing or state of affairs then perhaps also we should allow for conflicting and competing believings good that also track individual aspects – believing that the desired object has a certain good quality (the very quality in virtue of which it is desired). With these considerations in mind, there may be no clear and indisputable case in which desiring and believing good come cleanly apart.
If the mind works by the manipulation of discrete representations with discrete functional roles – inner sentences, say, in the language of thought, with specific linguistic contents – then the desire that P and the belief that P would be good are surely different representational states, despite whatever difficulty there may be in prizing them apart. (Perhaps they’re closely causally related.) But if the best ontology of belief and desire, as I think, treats as basic the dispositional profiles associated with those states – that is, if mental states are best individuated in terms of how the people possessing those states are prone to act and react in various situations – and if dispositional profiles can overlap and be partly fulfilled, then there may be no sharp distinction between the desire that P and the belief that P would be good. The person who believes that Obama’s winning would be good and the person who wants Obama to win act and react – behaviorally, cognitively, emotionally – very similarly: Their dispositional profiles are much the same. The patterns of action and reaction characteristic of the two states largely overlap, even if they don’t do so completely.
This point of view casts in a very different light a variety of issues in philosophy of mind and action, such as the debate about whether beliefs can, by themselves, motivate action or whether they must be accompanied by desires; characterizations of belief and desire as having neatly different "directions of fit"; and functional architectures of the mind that turn centrally on the distinction between representations in the "belief box" and those in the "desire box".