I do not have any sources in front of me, nor the availability to consult any sources at the moment, so what I have to offer is less an authoritative thesis than it is an understanding of my own making.

First, you've half-answered your own question. You say that both Manifest Destiny and Lebensraum involve the settlement of land around the United States and Germany, respectively--this is essentially true. But think about the land around each country. While Germany is bordered on all sides by rival states with ancient claims to their lands, the American West was seen as "virgin land"; nobody owned the West before American settlement (or, at least, that's what Americans convinced themselves of). After all, the earliest Americans--that is, the Puritans and the Jamestown colonists--demonstrated this by settling on "Indian land" when they reached the New World. Granted, the Indians did not have an European concept of land ownership, but even if they had, that likely would've made little difference to the settlers.

Next, we must address the justifications of each doctrine. Lebensraum was, like a great many German things, utterly pragmatic in nature. Lebensraum was motivated by a desire to strengthen the German Empire, and the surest way to do so at the time of its existence was to increase the nation's population through agricultural expansion and the nation's economy through industrial development. More German land meant more logging, mining, and farming, which meant a stronger Germany with more weight behind it on the international stage.

Manifest Destiny, on the other hand, was far more metaphysical in its design. Yes, the settlement of the West made America more profitable and powerful. But it's important to remember that the height of westward expansion in America was seen in the early-to-mid-1800s, while the German Empire would only pursue territorial expansion in the late 19th century until the beginning of the Second World War. Further, the German state was much smaller than the vast United States, meaning that German settlement was far more centrally-organized than American pioneering. While the products of newly-acquired German lands might contribute almost immediately to the German war machine/manufacturing industry/public works sector, the rewards in the American West would take years to make their way across the country, economically speaking. In light of this, we arrive back to the idea that Manifest Destiny was something far more ideological than Lebensraum.

Manifest Destiny was seen as, well, America's destiny. Note that "destiny" is, itself, a very spiritual word. It implies a degree of Providential guidance in whatever undertaking the subject is engaged in. To say that something "will happen" is far less charged than to say that something is "destined" to happen. And such was the belief of many Americans in the early nineteenth century, leading to our first point on Manifest Destiny's nature: its spiritual background.

The common theme between the many divergent churches which separated from the original Puritans in the Massachusetts Bay Colony was evangelism, the Christian practice of zealously spreading the faith. Add to this missionary genealogy the First and Second Great Awakening, and you have an American populace that, by and large, is hell-bent (no pun intended) on seeing every corner of the Earth baptized. Thus, the spiritual nature of Manifest Destiny: settling the rest of the continent was not seen as a patriotic goal, but a religious one--the settlers were not just spreading American civilization, they were expanding Christendom itself.

John Gast's American Progress is a closely-associated image with Manifest Destiny, and it exemplifies this point. The far corners of the painting, where the Indians and buffalo are, are dark and painted in drab colors. Meanwhile, the giant personification of America, Columbia, is dressed in white, and light follows her. On a shallow level, this is the light of civilization--she's spooling out telegraph wire as she progresses--but in the days before nationalism, hers is the light of Christianity.

I mentioned the "civilizing" nature of Manifest Destiny just now, and it'd be folly not to expand on this. Going back to my previous point, that the West was virgin land in the eyes of the settlers, we must consider what amounts to a sort of anthropology in Manifest Destiny. Europe, the Old World, was stained by history. There had been great and terrible empires across Europe for thousands of years, and they had left the continent with its share of political strife by the year 1800. America, the New World, was the "last, best hope" for human civilization. Since the land was unclaimed and unsettled, there was a belief that America, the consummate New Jerusalem, had the responsibility to export its greatness across the continent. In doing so, America would redeem the Old World and build the example of a perfect civilization.

Now, let us return to Lebensraum, because there are areas, frightening as they are, where it approached Manifest Destiny. Manifest Destiny was driven, in large part, by early American exceptionalism--the belief that Americans were solely unique, and, thus, superior, by virtue of their pioneering spirit, their piety, and their industry. Lebensraum under the German Empire and, later, the Third Reich, was similar to Manifest Destiny (as well as the Italian spazio vitale and Japanese *Hakkō ichiu) in that it asserted the inherent superiority of the German people. Thankfully, American exceptionalism did not make as heavy use of racial science as Lebensraum.

Where they finally diverge again is in their degrees of militarism, so to speak. Lebensraum was old-fashioned European conquest, in the same vein as Napoleon's empire and the Romans of old. It was the state-sponsored invasion and capture of foreign lands for domestic use, plain and simple. On the other hand, while the Indian Wars were prosecuted by the American government and its military, Manifest Destiny was rhetorically far from conquest. Circling back to the Indian notion of property, if the land was not owned, it couldn't be conquered, merely settled. Furthermore, because America was "distinctly" democratic, it's expansion was viewed as more than a patriotic cause--it was an independently noble one.

It is also worth noting that much of what contemporary and modern thinkers believe about Manifest Destiny originated with the Monroe Doctrine--the demand made by President James Monroe in 1823 that no European powers continue colonizing the New World. Westward expansion through Manifest Destiny not only assured America's future, but rebuffed the European empires, in line with our aforementioned "redeeming" of the Old World.

Now, practically speaking, both doctrines had, or would have had, similar effects. Westward expansion resulted in America becoming first a regional power, then a global one. Lebensraum, meanwhile, accelerated Germany's rise to become another major power in the world, on par with the much older and more famous British Empire. They also both involved the displacement and death of a great many indigenous peoples. In the end, however, there is far less objection to Manifest Destiny than Lebensraum, because while Lebensraum was a nationalistic policy which eventually devastated Germany and much of Europe, Manifest Destiny was an economic and spiritual advancement of the American cause, at no cost to the American people.

Follow-up question: I was told that Herman Lebensrau was inspired by the American Manifest Destiny. Can anyone confirm or deny and possibly elaborate?