Experimental Theology is actually a phrase I picked up from reading Phillip Pullman's The Golden Compass, but it expresses something that I think is one of the most important elements and key differences between the fullness of the Gospel and other religions. It is the idea that we can and should gain spiritual knowledge for ourselves. That is not to say that other churches do not preach this - just not with the same vigor and depth.
Faith for us is always based on knowledge of some kind - it is not meant to be completely blind. And to the degree we operating on blind faith, it is our duty to gain a testimony for ourselves. Faith does not exist just for itself, but as a step to learning or accomplishing something.
By no means do I mean physical proofs - like people who get caught up trying to prove the existence of God. While those proofs do occasionally exist (miracles, angels, etc), they are not what actually makes any difference for people. It is not what converts them.
When missionaries teach people the Gospel, invariably the first commitment they ask people to make is to read and pray about the Book of Mormon. Likewise we spend a lot of time in church talking about gaining a greater testimony. This is nothing to take lightly, even for old-timers. Sometimes there are those with "shaken faith syndrome" which happens when we have over estimated our own testimony - easy to do! The first time I delved into the details behind the Book of Abraham, I was confused and disturbed for several days - until i had time to learn and examine the details. This takes effort on our part!
But how can we most truly know something? It does not take watching many murder mysteries to realize that facts and details, can and often do obscure the truth. This is just as true in Science, as it is anything else. Ponder the importance and art of Advertising if you suspect I err!
The Holy Spirit is the spirit of Truth. Truth does not merely come from Christ - he IS the TRUTH and he is the WORD. Strong terms. Truth in it most pure and elemental comes only by the Spirit of God - not just our own opinions!
Further, it is a tenet of our faith that we are saved no faster than we gain knowledge. Likewise, that knowledge and intelligence are among the most fundamental aspects of God. In this sense, doubt is a spiritual pang, like a hunger pang, that tells us we have work to do - that we must learn something for ourselves.
If we are not coming to know the truth for ourselves, then we are not coming to Christ and if we are not helping others to learn the truth for themselves, we are not bringing others to Christ. Truth is available to us. It is not availability that is the great obstruction, but our own willingness to seek after truth, and to accept the truth once we receive it.
He has given us minds, spiritual and physical and he expects us to use both.
In the pre-existence, we operated with complete knowledge and we proved faithful when getting those details was easy. Now in this life, getting all the answers is hard, and sometimes impossible. There are questions to which I only have incomplete answers at best. But in my trials and problems, I have come to know God. In knowing Him, I trust him. Of course, since I work in the hard sciences, unanswered questions are a matter of course. And unanswered questions, like car accidents, are not a matter of 'if', they are a matter of 'when'. If Heavenly Father does not give us the answer in the detail we desire, then we must let our actions hinge on what else he has said.
If I am unsure of the Prophet's words, sometimes what I need most is not a testimony of that exact principle, but a testimony of the divine inspiration and call of Prophet. The same may be said of the Scriptures.
This is why the basics are so emphasized at Church - we need to read the Scriptures, pray, and keep the commandments so that we can have the Spirit teach us the things we need to learn. We need to seek after it... and sometimes even sweat a bit! And most of all, we need to be willing to receive what He tells us.