Before anything else, a point of order. This question has the ability to be construed as mean-spirited towards my Mormon brothers. I'd like to hope that I have the credibility on this site to be believed when I say, this question is NOT intended as a forum in which to bash Mormons. Answers that are not respectful to Mormons will be downvoted by me and will NOT be accepted.
So, when I was with Baptist missionaries in Kazakhstan, I first heard the line that "You know, Mormons and Muslims are almost exactly the same religion." They both claim that the Bible needed updating, they both deny the Trinity, they both tend to emphasize works over faith, they both are revelations by an Angel to a conveniently placed prophet, they both divide heaven into different levels for increasing belief, etc... And, recently I watched this 6 minute video from a Lutheran making a very similar claim. To an evangelical or another Christian predisposed towards rejecting Mormon claims, it's very persuasive stuff.
As such, I'd like to hear the counter-claim. How does the Mormon church react to being so closely compared to the Muslim religion? My suspicion, of course, is that the Mormon church rejects such comparisons, but I wonder what aspects of the faith they would use to differentiate themselves from Islam. Likewise, the video linked above is short enough to be fact-checked and debunked. Lutheran Satire is normally fairly level-headed (although deliciously funny, such as when they skewer Westboro Baptist Church). So, what is the case that Islam and Mormons have nothing in common?