General comparisons
The following comparisons were drawn:
- Charismatic prophet/spiritual leader (depending on what we mean by charismatic prophet or spiritual leader, this could also be applied to John the Baptist, Joseph ben Jacob, Abraham, Moses, Joshua, Gideon, Jeremiah, Nehemiah, Ezra, Peter, Paul, James the Just, and most importantly, Jesus. Many non-Biblical examples could be cited as well)
- Claims pre-existing scripture is corrupt ("incomplete" or "misunderstood" might be more precise terms) (consider also Moses, Isaiah, John the Baptist, and most importantly, Jesus). It is a hallmark of Biblical prophets & apostles that they correct the people for misunderstanding what God has said before and what He's saying now.
- Receives angelic ministration (consider also Abraham, Moses, Isaiah, John the Baptist, Peter, Paul, and most importantly, Jesus)
- Provides additional scripture (consider also Moses, Isaiah, John the Baptist, Matthew, John, Paul, James, Peter, etc, etc, etc. Most importantly, Jesus)
- Attracts early followers & implement works that encourage faith (consider also Moses, John the Baptist, Peter, Paul, and most importantly, Jesus)
- Becomes a prophet-king/civil & religious leader (this one is a stretch, see below) (consider also Moses, Joshua, Gideon, Nephi, King Benjamin)
- At the time of death leaves the movement in a healthy enough place that it can go on (consider also Moses, John the Baptist, and most importantly, Jesus)
Note the conspicuous presence of Moses in every category. John the Baptist makes all but one. Jesus Himself can be described in many of these ways. The comparisons suggested are general enough that most could be applied to almost any successful religious movement.
A few of the many noteworthy distinctions
How it spread
Islam spread rapidly through forcible conquest; The Church of Jesus Christ of Latter-day Saints came from incredibly humble beginnings and spread rapidly through missionary work (the latter is much, much more comparable to work of the 1st century apostles). Joseph Smith died an outnumbered, persecuted martyr; Muhammad did not.
A final word, or words that never cease
The Church of Jesus Christ of Latter-day Saints believes in ongoing general revelation and an open canon of scripture. Whereas Muhammad is thought of by Muslims as the great and final prophet, Latter-day Saints believe in a restoration of prophetic succession, and that there is a prophet today holding all the priesthood authority that Joseph Smith did.
A restoration
The Church of Jesus Christ of Latter-day Saints has a very distinct set of beliefs about apostasy & restoration, and see Joseph Smith as one of many in a line of leaders of dispensations (see here).
Numerous other distinctions in beliefs (some of them extremely significant) could be cited, but the question specifically seeks to understand origins.
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Explanation of the faith
The Church of Jesus Christ of Latter-day Saints believes that the fulness of of the Gospel of Jesus Christ was restored--by Jesus--through a prophet in modern times, and that all people are welcome & invited to test the truth of these claims and enjoy the blessings that come from them. We claim a set of beliefs rooted in personal revelation--that one can know for oneself, through direct revelation from God, that these principles are true.
I do not claim an in-depth understanding of the origins of Muhammad's teachings or intentions, but I cannot say I've ever encountered a set of beliefs or testimony in Islam comparable to what is described in the previous paragraph.
Appendix
Prophet/king
Is being a civil & religious leader a strong comparison between Muhammad and Joseph Smith? I doubt it. A comparison between Moses & Muhammad would be much closer.
The highest political office ever held by Joseph Smith was that of a mayor (answerable to a governor, Federal government, etc). Stronger comparisons might be made to Brigham Young, who was a governor, or Reed Smoot, who was a Senator.
If running for President of the United States is comparable to being a king, the United States has had an awful lot of kings.
Status of the movement when a leader dies
The question was asked whether the 1st-century church was in a good place on Good Friday. A similar question could presumably be asked about John the Baptist's followers when he died. I don't know that I would say they were in a good place, but they were certainly poised for great growth--this was my take on the OP's reference to the movement being in a healthy place.
Likewise, when Joseph Smith died in June of 1844, The Church of Jesus Christ of Latter-day Saints was not necessarily in a good place. They were under regular threats of violence from their neighbors (some of which threats were carried out). Within a short time they would be driven from their homes (some of them for the 3rd+ time), lose nearly all of their worldly possessions, and be driven from the United States (the Salt Lake Valley was not part of the United States in 1847). Thousands would die in the process. They were poised for great growth, but I wouldn't say they were in a good place.
The political & economic situation of The Church of Jesus Christ of Latter-day Saints in 1844 looks a lot more like Good Friday than it does the 7th-century Arabian peninsula.
Incomplete/corrupted scripture
I offered a more lengthy discussion here. I do believe that many "plain and precious things" (see 1 Nephi 13:28) were lost (certainly nowhere near all truths though!) from what was known by the prophets & apostles of prior eras--hence my comment that the Biblical record was "incomplete" - it neither contained everything God had ever said, nor everything He intended to say.
The Bible was & is, nonetheless, as the Book of Mormon itself states in the same chapter just cited, "of great worth" (see 1 Nephi 13:23). Indeed, the prophet Mormon said the Book of Mormon was written so that people would believe the Bible (see Mormon 7:9).
