The Deeper Difference
Rather than regurgitate another civics lesson about the democracy and the rule of law, I want to focus on a deeper difference between the abuse of prisoners and the beheading of Nicholas Berg. As far as I can tell, no soldier at Abu Ghraib ever shouted the Lord's Prayer while hitting a prisoner. No soldier ever justified his or her behavior by appealing to Moses or David or Jesus or Peter. I don't believe any soldier cried out John 3:16 while striking anyone or claimed he was defending Christendom (a word we almost never use anymore).
Contrast that with the video of Berg's murder and the text on the website that hosted it. The text itself mentioned "Allah" at least 11 times, "Islam" and "Muslim" 8 times each, and quoted the Koran at least once, "O you the prophet incited the believers to the fight." (I think that is 8:65 also translated as "O Prophet! urge the believers to war".)
Contrast that to the five hooded men on the video screaming, not yelling or shouting but screaming, "Allahu Akbar" while they sawed into Berg's neck.
That is the difference.