What are the differences between Mormonism and Islam, and how do Mormons react to comparisons with it? 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?
From an LDS perspective, Islam actually looks much more like traditional Christianity than Mormonism, because, despite the similarities mentioned above, it lacks what Latter-Day Saints consider one of the most important features of their faith: continuing revelation.

The simple fact of the matter is that Islamic doctrine ends with its "conveniently placed prophet," whereas Mormonism begins there.  Islam, much like conventional Christianity, has a closed canon and a belief that, after a certain arbitrary point in the distant past, no more prophets or scriptural revelation are given, or have been needed.

By contrast, the modern successors to Joseph Smith and the Apostles who presided over the church in his time are considered to have the same authority and divine responsibility today as their mid-19th century counterparts.  In fact, Wilford Woodruff, the fourth President of the Church, proclaimed that this is an essential feature of Mormonism:


  It matters not who lives or who dies, or who is called to lead this Church, they have got to lead it by the inspiration of Almighty God. If they do not do it that way, they cannot do it at all.
  
  -- Addendum to the first Official Declaration, emphasis added.


The importance of continuing revelation and the continuity of the authority of the church is also underscored in four of the thirteen Articles of Faith, which Joseph Smith wrote as a brief synopsis of the fundamental principles of LDS doctrine:


  5 We believe that a man must be called of God, by prophecy, and by the laying on of hands by those who are in authority, to preach the Gospel and administer in the ordinances thereof.
  
  6 We believe in the same organization that existed in the Primitive Church, namely, apostles, prophets, pastors, teachers, evangelists, and so forth.
  
  7 We believe in the gift of tongues, prophecy, revelation, visions, healing, interpretation of tongues, and so forth.
  
  ...
  
  9 We believe all that God has revealed, all that He does now reveal, and we believe that He will yet reveal many great and important things pertaining to the Kingdom of God.


While many superficial similarities exist between Mormonism and Islam (or, to be honest, between just about any two religions you could name,) the idea of continuing revelation, that the heavens are not sealed, that prophecies and divine leadership and guidance from the Lord have not ceased, is quite a unique thing, particularly among the Abrahamic religions.