The spiritual did not come first, but the natural, and after that the spiritual.
— 1 Corinthians 15:46 (NIV)

The prevalence of the belief in immortal souls or some sort of taken for granted eternal existence seems to colour most of our assumptions regarding the nature of Souls and Spirits.
I believe that Souls are abstractions for living body-minds (both mind and body together are souls). And Spirit is instead something that you develop (or is poured into you, so to speak).
Traditional Hebrew understandings of soul was that it was the same as the body, but the spirit was something else (Nephesh or soul, and Ruach, or spirit). Islam likewise distinguishes between Soul and Spirit (the Naf, or psychic organs, and the Ruh, or God-given Spirit).
Folk religions and non Abrahamic ones have similar ideas. Germanic Religion had four to five souls (Hamr or skin, Hugr or thought, Fylgja or astral familiar born into the world with a child, Hamingja or name/luck, and maybe Alf— literally elves or psychic organs).Chinese religion had at least two souls (Hun— the ethereal soul, and Po— the animal soul), though nowadays, there’s at least 10 souls in Chinese folk religion.
And we know all about the Egyptian varieties.
This brings me to Buddhism. There are five aggregates— form/rupa, sensations/vedana, perceptions/ samjna), mental formations/sankhara, and consciousness/vijanna — which functions in the same manner as as these orthodox understandings of soul as physio-psychic organs.
However, through spiritual cultivation, you can develop your Buddha Embryo (an abstraction for the potential of enlightenment inherent in all sentient beings, if not all phenonmena) into a Buddha-dhatu (Buddha-Nature).
What this is really saying, once you strip away the fancy terms, is that none of us have an immortal soul, but all of us have the potential to DEVELOP a Spirit.
Where do this Spirit come from? According to some, the Holy Spirit is God’s divine energy, pervading all phenomena that we are trying to open ourselves to. In Buddhism, the same concept is there— to become a Buddha is to realize Dharmadhatu (the pervading whole; or Dharma-Nature) and develop a Dharmakaya (The Truth Body). To transform this ordinary phenomena that we are made of into something more.
What do you believe?