There honestly isn't a very great difference between the two but the nuances exist. Off the top of my head:

• both were propped up by social Darwinist rationale that basically amounted to "We are inherently superior and have the ability and means to bring this land under our control and cultivation, therefore we should and will do so"

• both had this niggling idea that in order for society to progress it needed a balance between industrialization and agrarianism. Germany during WWI saw massive food shortages so the rationale behind Lebensraum was very easy to understand for civilians and veterans who had managed to survive the threat of starvation not just once but twice (WWI and Great Depression). Anything that would assure a steady supply of food was surely a good plan to these people

• Adding to that, there is an emphasis on the inherent superiority of a particular "race" of people as being both the dynamic force behind it and the successors --for America it was the "Anglo-Saxons"; for Hitler it was the Aryans, but particularly the ideal German people.

• On the other hand, while Nazi Germany was a one-party state that eliminated any organized political dissent from opposing parties that would have protested something like Lebensraum, some key figures did in fact object to the aggressive expansion in the West. Most Whig party members were against Manifest Destiny because 1) there was the worry that slavery would spread to those new states, and 2) they were much more interested and geared towards developing and industrializing the land already under their control. Several Presidents such as Abraham Lincoln, Ulysses S. Grant, and John Quincy Adams were against Manifest Destiny for reason #1; Grant especially was of a mind that the Native people should be peacefully integrated into Western civilization and made to settle down. While this was greatly ignorant of their culture and lifestyle, he still made efforts to avoid confronting natives with bloodshed (it still happened, like with Custer, but you can't have everything)

• Further justification for Manifest Destiny was basically "We're doing these savages a favor. Once everything is farm land and actual civilization is built up, then we'll all real the benefits from using this land properly." Justification for Lebensraum ideologically was that the people occupying the lands in the East were, to put it in bigoted terms, inferior races that threatened the purity of the German people and therefore were an obstacle to be overcome. As such, troops went out of their way to clear whole towns and villages of their inhabitants for native Germans to be transplanted into unless those inhabitants could be proven to be "ethnically German". In a way, Lebensraum was an offensive measure as a defensive measure against the perceived threat of Slavs and Jews, a way of eradicating an inferior race. By contrast, Manifest Destiny, while still containing undeniable racism, had more underlying ideas of "white man's burden" to cultivate land that the natives had not "used correctly" and to either re-educate and assimilate them into the white man's way of life.

I was watching School House Rock's "Elbow Room" with my 10 year old, and the 16 year old, who had just watched a WWII special, asked this question.

Black Earth by Yale historian Timothy Snyder contains, among other things, a very good discussion of Nazi and particularly Hitler's attitudes toward's lebensraum. What the book does particularly well is to locate lebensraum within the geopolitics of eastern Europe, especially nationalist movements in such places as the Ukraine, at that time.

Those geopolitics, I would argue, are one inherent difference between the two. Another is that, at least Snyder would seem to argue, lebensraum was a more cohesive product of Nazi ideology and particularly Hitler's way of viewing the world, whereas there was no one proponent of manifest destiny that both expressed a comprehensive, agreed-upon list of objectives in the way that Hitler and his small Nazi circle did.

Hitler was strongly influenced by a concept Snyder terms "judeobolshevism," the equation of Soviet power with Jewish influence. One tenet of this concept was both Jews and Communists could not defeat stronger races (i.e. Germans) in a straight fight, and so instead fought using starvation. Hitler, for example, characterized Allied blockades (which, being Hitler, he attributed ultimately to Jewish power) in WWI as starvation weapons. Stalin, in the years leading up to the war, of course did actually use starvation as a potent weapon against nationalist movements and other perceived enemies in such places as the Ukraine. (The Holodomor would be an example of this.) So Lebensraum was not envisioned solely as a way to make Germany greater (it might be more precise to say "make Germans greater," actually, as Hitler tended to think in terms of races and not states), but rather as a bulwark against a racial war of starvation he convinced himself was coming.

While both Lebensraum and Manifest Destiny played on an inherent superiority- it was uniquely the white American's destiny to settle North America, bring civilization to the wild regions and bring all it's trappings with it- Lebensraum differs in the sense that it had genocide baked into it. Manifest Destiny wasn't nearly as codified in documents, nor did it actively advocate for the extermination of 50-99% of all ethnic groups west of the Mississippi, where as Lebensraum absolutely did. And when I say that Manifest Destiny wasn't really codified, I mean at all. It was a vague concept as opposed to official government documents.

More over, Lebensraum was pretty overt about it's intentions with Eastern Europe- colonization. It was extractive in nature- framing WW2 as the 'end game' of the colonial period is perfectly valid- where as Manifest Destiny could be interpreted more as bringing something. It's rather difficult to plan an extractive, colonial effort on a region where you can barely even bring people, where as Hitler was operating with very solid ideas of what the land he wanted contained. Regions like Ukraine were singled out not because it'd be a chill place to live, but because it was a major breadbasket.