This is not a merely linguistic question since desire and motivation fulfil logically different roles in the explanation of action.
Desire connects with motivation at least in this way: If I did an intentional action, say I bought a packet of cigarettes without coercion or constraint, then it makes sense to ask what my motivation was in buying the cigarettes. Didn't I realise the health risks associated with smoking ? Why did I buy the cigarettes? When you have answered the 'why?' question, then you know my motivation. A standard model, going back at least to Hume, has it that the motivation for my intentional action involves a belief and a desire. I bought the cigarettes because I believed they were available and because I desired to smoke them. On this approach, desire along with belief is one of the elements that explain why I did a particular intentional action, in other words a specific desire constitutes one part or element of my motivation where a specific belief constitutes the other. If desire constitutes a part or element of motivation then they are on different logical levels.
It isn't the case, though, that desires invariably motivate. I can have a desire to smoke a cigarette and yet repudiate that desire. I have a desire to smoke but also a stronger desire not to smoke. If a desire can be present without a motivation then again they are logically distinct.
Desires can also fail to form and hence to motivate if a certain belief is absent. If I believe that the building in which I am working is on fire, then I am very likely to have a fire-avoiding desire to quit the building. However, the building can be blazing within inches of my office yet I will have no fire-avoiding desire to quit the building if I have no belief that the building is on fire. One way of describing this kind of case is in terms of a distinction between dispositional and occurrent desires. I have, let us asssume, a dispositional desire to quit burning buildings - a prospensity to form desires to quit such buildings - but there will be no occurrent desire, no desire to quit this building now, unless I believe (as I may well not) that the building is on fire.