I find the above answers to your question unsatisfying in one way or another. (Not that I have a more conclusive answer hiding up my sleeve for you.)
Nonetheless, contrary to some earlier responses, it seems a very pertinent philosophical question to me. (Of course, if you know that you already know all that there is to know about philosophy, then you might be in a position to definitively rule in and rule out what is and isn't a relevant question for everyone and anyone else at any time. However, such a person would seem to me to be better described as a sage, not a philosopher)
CesarGon's attempts to delineate between the two terms seem a bit too reductive. To say that someone digging a hole at gunpoint is not acting on a desire seems very unsure. They would surely be acting under the very palpable desire of "not getting shot" or of "staying alive." Therefore CG's attempt to separate that state analytically from other (more positive) states of desire seems inadequate.
Similarly, CG defining a motivation as something that "compels us to do something", hardly seems to separate it from a desire. In fact one could argue the exact opposite: that a subject is more compelled by the state we call desire than by a motivation per se. I might be motivated to find a new car... but, as any advertiser knows, that motivation may always remain latent unless I can be taken hold of by a particular desire to buy this car in front of me.
This latter example seems to me to bring us a little closer to the defining line (at least in use) between these two terms:
Motivation often appears to convey a more neutral (de-subjectivised) narration/explanation of possible reasons for the appearance of an action or event. Motives in this sense are plural and interchangeable. One can be moved to an action on the back of multiple motives (ie. they are plural). And one can share all these motives with vast swathes of the greater population. I do my exams because I am motivated to succeed, to please my parents, to earn money, etc. One can make a list of one motivations and cumulatively keep adding to the list in order to keep oneself motivated (i.e. they are interchangeable).
Desire, on the other hand, seems the much more elusive and subjective term. One is generally not speaking neutrally when one attributes the term desire to a particular situation or state. Even more strangely, when compared to motivation, it appears both the more individual and the more universal term at the same time.
In one sense, a desire can be understood as the particular individuated (subjectively experienced) state of being motivated. But this seems to say too little. Any motivation would in that sense be a desire when experienced subjectively. To some extent this is correct.
But very often we mean something different, something more fundamental, when we speak of desire. In this sense it relates not to a multiplicity of motivations, but to some ur-motivation; some central motivation that drives our other motives (from the start?) and directs them. "What is it you desire?" This sense of the term has a more metaphysical ring to it, which is why it has long been a central concern of philosophy in one disguised form or other (from "conatus" to "will" to "the unconscious" and on and on).
To return to our car-purchaser analogy: the purchaser may be motivated to buy a car because they need a form of transport. But this hardly seems to explain the extremely prominent place of automobiles in modern society. A broader explanation might posit that the need to replace one's car, with a newer "flashier" car every year, is motivated by the wish to compete in a hyper-materialist and strongly hierarchical social world, where image has become increasingly important. Positing an ur-motivation like this brings us closer to the term "desire" as it is normally used. It would seem to indicate a more ultimate/ulterior motive: what a subject "really" desires. For example, one might undercut even the latter explanation of car-purchasing by positing a still more expansive/original explanation: that the "motivation" to buy a new car is not simply about people wanting to "keep up with the Joneses", but is itself sparked by (and absolutely necessitated by) the far more fundamental desire of the Capitalist productive structure itself to always produce profit and economic growth (i.e. a fundamental socio-political motivation completely indifferent to the subject's individual motives). Desire in this sense influences, drives and perhaps even determines other more mundane "motivations" that we might point to as explanations for our behaviour.
For this reason, desire (as ur-motivation) often appears as something universal... but at the same time, something less plural or interchangeable. No doubt we do contain multiple desires. But, by comparison with motivations, one often experiences acute conflict when one experiences multiple desires. One could be motivated to be either a doctor or a painter and not be too worried either way. However, if one truly desires to be both, then one might feel acute conflict at choosing between the two life-paths... and/or experience an intense sense of lack or loss when one finds that they cannot fulfill one or other of these two desires.
The much more obvious example here would be the interconnections between the terms desire and love, or desire and lust. This connection does not hold for "motivation". You would not produce a neutral list of specific motives/motivations when asked by your partner/lover "why do you want to be with me?". Producing a piece of paper with a list of neutral and cumulative motivations (such as "living together keeps the rental costs down," or "I wish to have a live-in lover", etc) is likely to bring a quick end to one's relationship. When we are asked such questions (or when we ask our partners), we wish to get a sense of the "ur-motivation" that drives our relationships. What is it that drives our relationship over and above the plurality of mundane motives that we experience alongside it? This is what desire normally refers to. If that desire is lost then, for many, no list of other motivations (no matter how long or detailed) can stand in as its replacement (i.e. desire is not interchangeable in the way motivations are).
Okay, apologies, that turned into an ad lib essay. Hope it helps someone in some way. I know this is a ghost thread by this stage. But your question helped me clarify some of the issues for myself at least!