This article explains it pretty well.
Normally the Church uses these words interchangeably, and we have scriptural proof for that. St. Thomas Aquinas did mention something about a spiritual soul and a non-spiritual soul, which you might find similar to what your Protestant friend is talking about. But St. Thomas only said there are different characteristics of the soul. He did not say that our soul (or soul+spirit) has different parts.
Although it is true that after receiving the Holy Spirit, we have a Spirit in addition to the human soul Personally I believe the Holy Spirit is free to act in anyone’s heart, so if you understand it that way, it is plausible.