So let me ask this. Why Mormons believe that Joseph Smith visions and Book of Mormon is any better ( or more believable ) then Bible, Quran, Torah, NWT and so on? Why would non religious person who is looking to chose one of the religions should believe Book of Mormon and not the Quran for example? Why Mormons claim that Muhhamad is not a prophet ( I agree with them here) of God but Joseph Smith is?
That's a complicated question, and there are likely as many answers as there are Mormons. Just to clarify one point, though, Mormons also are Christian and believe in the Bible.
For me, I have read the Book of Mormon many, many times, and believe it comes from God. I do not believe that a man could have created such a book, especially not an uneducated 24 year old farmer like Joseph Smith. Its teachings, and the power I feel as I read it, affect my life in many ways and make me a better person. The Bible does that as well.
As to why I have a belief in God to begin with, I have had many spiritual experiences where I have felt a power beyond my own. It is consistent with the testimony of the prophets in the Bible, Book of Mormon, and modern day ones, namely Joseph Smith and his successors. As I pray, I have felt answers to prayers. Believers will attribute this to the Holy Spirit (aka Holy Ghost); unbelievers will say it is my own imaginings. Some of these answers to prayers have led me to believe that Jesus Christ is my savior, and that he died to provide an atonement, or reconciliation between us and God. (That rules out Muhammad's teachings for me, by the way, because he taught that Jesus was a prophet but not anything more than a man.)
If you really want to understand this, I recommend you try reading the scriptures such as the Bible and Book of Mormon yourself, and pray to see if you feel the Holy Spirit in response. Once you feel God working in your life, you can then decide which path He wants you to go down.