If you read Genesis 3, carefully, you will understand what "spiritual death" is all about.
As JThomp42 said, spiritual death is separation from God. I've taken this thought a bit further and discovered that this "separation" is itself called "ego."
Just look at the idea of what "ego" means. It is all "me, me, me!" Separate from everyone else. Ego is "I'm right and you're wrong." Ego is the sense of entitlement that so many in modern society wear like a badge of honor.
Physical death is a separation of spirit from the physical body, plus a cessation of bodily functions.
JThomp42 is right about the similarities between the two kinds of death. But there's more to this idea that needs explanation. Both kinds of death are also a cessation of consciousness. In the Garden of Eden, we found that our spiritual consciousness ended when we took the forbidden fruit.
With bodily death, physical consciousness ends.
The "Everlasting Life" of which Jesus spoke is the restoration of spiritual consciousness. This is the immortal spirit (child of God, true self, soul, Holy Ghost) reawakened or reborn.
What makes this particularly exciting is that when body death occurs while the spirit is fully awake, consciousness does not end, because the consciousness now resides in permanent and immortal spirit. This is the foundation that lasts. The physical consciousness is the impermanent quicksand of natural reality. It doesn't last.
So, what was the "forbidden fruit?" What was it that caused each of us to lose our connection to God and to lose our spiritual consciousness?
The chief clue is in the name of the tree -- knowledge of good and evil. It is a fruit, or product (result), of dichotomies -- action-reaction, generous-selfish, wisdom-stupidity, compassion-indifference, good-evil, and many more. These are the bone and blood of ego. This is why we have karma -- what Jesus called dying by the sword by which one has lived. This is an action-reaction (physical), false self.
So long as we live in ego, we are not following Christ.
I have seen what it's like to be without ego. My spiritual eyes were opened after millions of years. I weep with joy at that breakthrough. I could see the universe without physical (human) eyes. But then I became self-consciousness and darkness descended once again. Even so, I've had a taste of the everlasting life Christ spoke about. There is still hope, if I remain faithful.