Why do people die but angels do not, but both have sinned? In City of God, Book XIII, Ch. 1, Augustine wrote,


  Having disposed of the very difficult questions concerning the origin of our world and the beginning of the human race, the natural order requires that we now discuss the fall of the first man (we may say of the first men), and of the origin and propagation of human death.  For God had not made man like the angels, in such a condition that, even though they had sinned, they could none the more die.  He had so made them, that if they discharged the obligations of obedience, an angelic immortality and a blessed eternity might ensue, without the intervention of death; but if they disobeyed, death should be visited on them with just sentence—which, too, has been spoken to in the preceding book. 


It is my belief that only God is inherently immortal (1 Tim. 6:16), being the Creator, and all else, being created, and existing by Him (Acts 17:28), is granted immortality according to His grace. Thus, angels do not die like men, for they are granted immortality. Furthermore, man's soul/ spirit, is also granted immortality according to God's grace. Hence, upon death, while man's body perishes and endures corruption, his soul/ spirit continues existence.

But, I wonder, how is it that angels who have sinned (2 Pet. 2:4; Rev. 12:9) continue to exist in an immortal state (for it is certain that Satan sinned, yet he still exists, as the biblical authors attest), yet man dies as a result of Adam's sin? 

What is the explanation of this phenomenon?
Angels and humans are not of same essence. 

Humans are composed of physical bodies and immortal spirits.Humans experience human death.Human death is separation of  body and spirit.


  James 2:26 (NIV)
  
  As the body without the spirit is dead, so faith without deeds is
  dead.


On the other hand, angels are just immortal spirits without physical bodies. 


  Hebrews 1:7 (NIV)
  
  In speaking of the angels he says, "He makes his angels spirits, and
  his servants flames of fire."


The death that Adam and Eve experience could only be physical death (first death) and the second death ( in Gehenna i.e. Hell). 

In the second death, both sinners and the devils would experience the same kind of death.

Note that the second death ( in hell) if second death for humans and first death for angels.


  Revelation
  
  20:14 And death and hell were cast into the lake of fire. This is the
  second death.
  
  20:10 And the devil that deceived them was cast into the lake of fire
  and brimstone, where the beast and the false prophet are, and shall be
  tormented day and night for ever and ever.