There are lots of differences, but I think two of the main ones are: 
1) - The LDS church is a restoration of the church that Jesus established. With that there has been a restoration of the same priesthood Jesus Christ gave his apostles. If you think about it there are only two Christian churches that have claim on having the proper authority: the LDS church and the Catholic Church. 
 
2) The other thing that sets us apart in addition to the priesthood is the Gift of the Holy Ghost. With the Holy Ghost members of the church can receive a very real, spiritual witness from our Heavenly Father that the things that we teach are true. 
 
2. How does anyone know? By faith and by a witness of the Holy Ghost. Our Heavenly Father wants us to come to earth, be testted, and live by faith. If we had physical evidence of every little thing - there would be no need for faith. 
 
It is my testimony that these things are true. It is up to everyone to pray about the church, Joseph Smith, the Book of Mormon, etc. and receive there own witness. If they do - they can receive it. 
 
James 1:4,5 
5 If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him.  
6 But let him ask in faith, nothing wavering. For he that wavereth is like a wave of the sea driven with the wind and tossed. 
 
Here is the promise from the Book of Mormon. Moroni 10:3-5: 
3 Behold, I would exhort you that when ye shall read these things, if it be wisdom in God that ye should read them, that ye would remember how merciful the Lord hath been unto the children of men, from the creation of Adam even down until the time that ye shall receive these things, and ponder it in your hearts.  
4 And when ye shall receive these things, I would exhort you that ye would ask God, the Eternal Father, in the name of Christ, if these things are not true; and if ye shall ask with a sincere heart, with real intent, having faith in Christ, he will manifest the truth of it unto you, by the power of the Holy Ghost.  
5 And by the power of the Holy Ghost ye may know the truth of all things.

The questions you are asking are questions I think everyone asks at some point in life. 
 
Actually, this is just the thing Joseph Smith was wondering at 14. He wrote (speaking of religion): 
"It was impossible for a person young as I was, and so unacquainted with men and things, to come to any certain conclusion who was right and who was wrong. . . . 
In the midst of this war of words and tumult of opinions, I often said to myself: What is to be done? Who of all these parties are right; or, are they all wrong together? If any one of them be right, which is it, and how shall I know it?" 
 
Joseph Smith later read James 1:5 
"If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him." 
 
He then decided "if any person needed wisdom from God, I did" 
 
So he decided to pray to find out which church to join, and to his surprise, God told him to join none of them. 
 
I think you know the rest of the story. 
 
Jett answered your previous questions, but I wanted to add Joseph Smith's account of how he wondered these same things about other religions. He prayed to find an answer, and we can too. 
 
You're right that "everyone experiences a time when they felt something in their hearts but were wrong about it" But the feeling we talk about in the heart, is a very specific feeling. It's the Holy Ghost, who bears of truth, and when he bears the truth, you gain a knowledge--not a intellectual knowledge, but a spiritual knowledge. The Holy Ghost communicates to your spirit. 
 
Learning how to discern the feeling of the Holy Ghost from your own feelings takes concious practice. The Holy Ghost is often a quiet feeling, or a thought that suddenly enters your mind on what seems like on its own accord. Sometimes the feeling is more obvious, like a pounding firy feeling in the chest, or he becomes audible to your physical ear. 
 
I believe the Holy Ghost is the witness of truth. I have felt like Joseph Smith felt at 14. Every religion has such different interpretations of the Bible, which is right? I've learned that people can argue about who is right and who is wrong and which interpretation of the Bible is right until the cows come home, and that's why the Holy Ghost is so important. You can only know the truth through the Holy Ghost. No one can find the truth for you. You have to find out for yourself. Think about it, Study it, Pray about it, and the Holy Ghost will witness to you what is true.