Seems like there are many differences in nuance, such as what one desires, whether it includes receptivity, intent, and expectation, and finally whether it includes longing- a deep sense of lack. 

Abraham-Hicks interpretation as I understand is that longing is not desire. Desire plus expectation creates the wanted manifestation, but that longing manifests increased lack.
 
My Chi Gong teachers tell me that craving [i. e. lust and greed] is unfulfilled desire and causes liver chi stagnation. Hence more craving. The Taoist philosophy and cure for this is to learn to be content, i.e. gratitude.
 
I think the bridge to this is that gratitude cannot be forced, but it can be cultivated. In my understanding,  Abraham-Hicks says that happiness comes naturally after being nourished with the intention of focusing on what is good in the present. The reason I like the work is that not only is there an explanation but also a corresponding technique. Taoism also encourages cultivation.
 
BTW, in Taoism unlike in some forms of Christianity, the desire for standard things like good health, spiritual growth, happy family [which includes a decent income and a sex life] are not longing but healthy desires. Obsession over any of these things, however, precludes the health part and moves into the craving category.
 
Chris