The difference in their estimate of total excess deaths (655,000 vs. 393,000) isn’t huge for a study with such inherent difficulties, but the difference in the violent death rate is. The Lancet study calculates that 92% of all post-invasion excess deaths were from violent causes, while WHO figures it at 38%.
Be all that as it may, The New York Times observes that “because of its timing, the study missed the period of what is believed to be the worst sectarian killings, during the latter half of 2006 and the first eight months of 2007.” The Lancet study, clearly, having been from even earlier also missed that period. So even if you have a great deal of confidence that one or the other of these studies got things right, the studies’ figures are probably badly outdated by now.
One issue this whole controversy raises relates to casualty figures from civil wars that aren’t political hot potatoes in the United States. When a study comes out in Iraq, there’s intense political pressure from one side or another (or both) to expose real and imagined methodological flaws. But what about things like the millions who’ve died recently in Congo where the studies get done in a context where nobody’s seriously trying to work the refs?