When God said to man that if he sinned he would die, was he speaking of death of the human body or death of the Soul, or the Spirit? When God warned Adam about eating from the tree of the knowledge of good and evil he said that in the day that he ate there of he would surely die.

Genesis 2:17  KJV


  But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die. 


We know that Adam lived over 900 years on the Earth:

Genesis 5:5  KJV


  And all the days that Adam lived were nine hundred and thirty years: and he died. 


So Adam and Eve did not die physically that day,  but  the argument could be made that they did die in the respect that they were now separated from God for eternity.

Since God had created man in their image, which I take to mean as in three parts body, spirit, and soul.

If sin had separated man from God eternally, and man was condemned to eternity in the lake of fire that would be the second death:

Revelation 21:8  KJV


  But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.


And since man was condemned to the lake of fire for eternity, which is the second death that would have taken place on the same day as their sin. 

Could that have been the reason why:

Genesis 3:22  KJV


  And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever:
It can't have been his body, because he was still alive physically and didn't drop dead on the spot.

And it can't have been his soul because he felt naked after eating of the forbidden fruit.  Which means his natural mind was still working.


  And they heard the voice of the LORD God walking in the garden in the
  cool of the day: and Adam and his wife hid themselves from the
  presence of the LORD God among the trees of the garden. - Gen 3:8


The only part left of us is our spirit which we cannot perceive through our 5 senses.  


  That which is born of the flesh is flesh; and that which is born of
  the Spirit is spirit. - John 3:6


So I believe it was a spiritual death, not a natural death.

Why was he driven out of the garden?  I believe it was so that he couldn't continue to eat of the tree of life and continue to live physically forever in a fallen state.


  And the LORD God said, Behold, the man is become as one of us, to know
  good and evil: and now, lest he put forth his hand, and take also of
  the tree of life, and eat, and live for ever: Gen 3:22


I believe it wasn't so much that sin had separated God from man, but man from God.
It was man who hid from God in the garden after eating the fruit and man who felt shame.  God continued to reach out to man but sin kept man away from God.  

But obviously things were not quite the same.  Without a spirit that is alive they were unable to fellowship with God.  They (and we) cannot relate to God spirit to spirit unless we are born again.

Gen 3:8-10:


  8 And they heard the voice of the Lord God walking in the garden in
  the cool of the day: and Adam and his wife hid themselves from the
  presence of the Lord God amongst the trees of the garden.
  
  9 And the Lord God called unto Adam, and said unto him, Where art
  thou?
  
  10 And he said, I heard thy voice in the garden, and I was afraid,
  because I was naked; and I hid myself.