According to a common view, beliefs suffer a coherence constraint that desires do not. If I believe that p, then I'm very unlikely, at the very same time, to believe that not-p -- and if I do, that's a clear rational failing. But desiring various contradictory things is commonplace.
I don't want to dispute that the English statement of the common view just given can often express a truth. But I think it's a mistake to infer from this that there's something interestingly different between the natures of belief and desire. The words 'belief' and 'desire' are used loosely to refer to a few different kinds of things. And the ones that are of most interest in a lot of philosophy, I think, are structurally much more similar than the common view would lead one to think.
Notice, for instance, that there's a way of talking about 'belief' that does not require the coherence constraint. The way in question is a less fully-committed one than the one that is required for knowledge, or explains action. I may be said to 'believe' that p merely by virtue of finding p rather likely. But of course, there are sets of propositions, each of which I (rationally) find rather likely, but which are logically inconsistent. There's another, more committal, sense of 'belief', which does seem to enjoy something like the coherence constraint. It's this one that figures most prominently in epistemology. It's also, I think, this stronger notion that is particularly interesting in action explanation. To act is to commit oneself.
What of desire? In ordinary, loose speech, 'desires' might not amount to much. I'm said to 'desire' a third pint merely by virtue of having a felt inclination towards it. In this weak sense, we often face conflicting desires. But this might not reach the level of commitment needed to explain a decision; to explain actions, we cite 'desire' in the stronger sense of an all-things-considered desire. ("If he'd really wanted another pint, he wouldn't have gone home.") In this sense, it is a rational failure -- and a reasonably unusual situation -- to have contradictory desires. Contradictory desires commit one to preferring an inconsistent situation, just as contradictory beliefs, in the strong sense, commit one to being in an inconsistent situation.
The two weak senses described don't seem to have all that much of interest in common with one another. (The belief-like analogue of the felt-inclination sense of 'desire' seems more like intuition than like the finding-likely sense of 'belief'.)
But the two stronger senses, I think, are interestingly related. They're the things, for instance, that figure into belief-desire explanations for action.
When we discuss 'boxological' approaches to cognitive architecture, it's 'belief' and 'desire' in these strong senses that constitute interesting boxes worth theorizing about.