When I was a missionary, we all taught that just as physical death was the separation of the spirit and body, spiritual death was a separation of our spirits from God, which we interpreted to be sin.
Jacob, however, seems to say that spiritual death is something else. He calls spiritual death hell.
O how great the goodness of our God, who prepareth a way for our escape from the grasp of this awful monster; yea, that monster, death and hell, which I call the death of the body, and also the death of the spirit. (2 Nephi 9:10)
Subsequent verses go on to describe hell as the place where spirits dwell as they await the resurrection. They also go on to describe the grave as physical death.
And because of the way of deliverance of our God, the Holy One of Israel, this death, of which I have spoken, which is the temporal, shall deliver up its dead; which death is the grave.
And this death of which I have spoken, which is the spiritual death, shall deliver up its dead; which spiritual death is hell; wherefore, death and hell must deliver up their dead…(vv. 11-12)
It seems then, that Jacob is saying that death is not the separation of the spirit and the body. Rather the spirit and the body separate to their own death: the body to the grave and the spirit to hell.