What then is this “spirit”? Are we conscious of being with God?
If not, what use is it to us, or God?
There is an ample amount of literature on this. If you're genuinely curious you can begin where the matter is addressed at the inspiration of God, the Bible. Here is but one example, from Hebrews 4:12:
For the word of God is quick, and powerful, and sharper than any twoedged sword, piercing even to the dividing asunder of soul and spirit, and of the joints and marrow, and is a discerner of the thoughts and intents of the heart.
The Apostle Paul wrote more than just a bit upon the soul and the spirit as well. I've previously mentioned that the soul is clearly in the blood, according to the Old Testament Leviticus. It's a misconception that the two, soul and spirit, are synonymous. A thorough reading of the Bible will bear this out. A look at the books of the Bible in the original language(s) will even further bear this out as well.
Soul and spirit are not the same. There is a natural body and a spiritual body, according again to Paul in I Corinthians. The soul is the life you live, your personality and your memories within that natural body.