What is the difference between Soul and Spirit?

Christian theology deals with God's relationship with us and God's work within our soul for our salvation. Christian theology leaves open to philosophical speculation the constitution of our soul and mind, as long as the resulting theories stay faithful to the theological anthropology taught by Scripture (see Wikipedia article and GotQuestions article on Christian anthropology).

In this answer I will correct some mistakes in your Question and provides a rough outline of the Thomistic Christian anthropological scheme which stays faithful to Scripture as well as time-tested. I will also briefly talk about how the Holy Spirit 1) gives us grace for faith necessary for conversion, 2) dwells in our soul, and 3) sanctifies our soul. I will also compare the Holy Spirit's work within our soul with the Holy Spirit's work in Jesus's human nature.

First, about our soul

1. The term "soul" refers to the immaterial part of our nature. Instead of the Cartesian notion that the soul is trapped in the body or the materialist's notion that the soul is an emergent property (which is even worse), Thomists teach that our mortal body is animated by our immortal soul; the body being the physical instrument of the soul. In contrast with the soul that animates animals, our soul is of a different kind: a rational soul made in the image of God possessing the CAPACITY to know and love God. While angels don't have bodies (they are solely immaterial souls), humans are hybrid (body and soul are intimately connected), which is why death feels so unnatural, until God gives us new glorified bodies so our souls can have EVEN BETTER instrument.

2. Soul is the seat of our consciousness, our thinking, our willing, our desiring, and our enjoying of the fruits of our act (the one doing reflection / contemplation). It is also in the soul that we become aware of our conscience (= "co-knower" of our acts), of God's presence, of God's divine and natural laws, of God's love and mercy, of God's nudging and guidance. For Thomas, "mind" is just another label for "soul"; he uses "mind" when referring to the soul as the one doing the thinking and the willing.

3. There are parts of our body that affect the soul such as our bodily desires and bodily emotions; the soul's job is to regulate them especially because those desires and emotions are unruly ("bent" by sin) due to the Fall. Please note that there are also intellectual desires which come directly from the immaterial soul; they include our desire for God manifested in our desire for Knowing (truth), Loving (goodness), and Appreciating (beauty). While animals simply follow their bodily desires to fulfill the ends God created for them, humans are designed to "have a say" in directing our bodily desires and in regulating our bodily emotions to achieve the ends that God has imprinted indelibly in our rational soul. Because intellectual desires are spiritual (like God), there is an affinity between 1) our desire for God, 2) our capacity to love and know God and 3) our responsibility to manage our bodies to make circumstantially appropriate decisions to fit the ultimate end that our minds are able to perceive by virtue of our rational soul, an ability that St. Thomas calls Prudence.

Alas, our minds are DARKENED by sin, so without divine help (described in the section below), our Prudence is insufficient for us to reach heaven. Thus, we are constantly trying and longing for God in BLINDNESS (substituting God for worldly goods) and when in our better moments we have a GLIMPSE of God we are frustrated by our inability to fully love Him (impotence), a glimpse which could have aided us to doing what SATISFIES us (beatific happiness as a result of acting virtuously). Thus without grace, we reach "dead ends" instead of heaven.

Then, about the Holy Spirit

1. In the Thomistic scheme (dualism), there is no human spirit, only body and soul as described above. Being made in the image of God, the soul has sufficient CAPACITY (potential) to act humanly (in contrast to animals governed only by passions) and to sense God imperfectly, but our soul needs to be elevated by grace to overcome the darkness of our soul/mind (that obscures our vision of God), which causes us NOT to seek God as our ultimate purpose / meaning as we should. Secondly, our soul/mind is dominated by sinful desires, sabotaging our attempts to do good. That is why Christian theology says that without the Spirit of God we are dead.

2. When we become Christians, the Holy Spirit dwells in our soul, communicating the grace to give us the light of faith (curing our blindness) that started small but is growing so we trust God more as our ultimate end, thus giving us the divine wisdom for our mind to make good choices that lead to life and happiness in God. Secondly, the Holy Spirit also communicates the grace that affects the will (curing our impotency), enabling us to love God and our neighbors more purely: with the agape love that is more tuned with God as our final end, rather than with natural loves whose ends remain at the level of worldly wisdom. In this way we gradually become LIKE God (fulfilling our rational soul's potential) as we PARTICIPATE more fully in God's divine life given to us.

3. It is very important to realize that the Holy Spirit does NOT supplant our soul, nor erase our individuality. He gives Himself to us, becoming the SPRING of spiritual life in our soul that in turn elevates, perfects, purifies, sanctifies, and empowers our soul. The Holy Spirit is free to come and go; we are also free to shoo the Holy Spirit away ! But if we are obedient and docile to the promptings of the Holy Spirit, we wouldn't want to shoo away the spring / source of life that we SEE making us more perfectly happy (by our mind's eye aided by the light of faith), even though we sometimes need to suffer other people's sins (whether for Christ or not) and even though we sometimes need to endure sickness, poverty, grieving, etc. By the light of faith we can chalk up those sufferings as necessary sanctification / purification work instead of regarding it as condemnation from God or as meaningless evil. As you pointed out, the Holy Spirit enables us to attach ourselves to Jesus, and Jesus (being God incarnate) will not let us go if we let Him dwell in our soul.

Finally, about Jesus's human nature and the Holy Spirit

1. Despite Jesus's human nature being exactly the same kind as ours (thus being the rational soul that works exactly the same way as ours), Jesus is FIRST Logos who upon virginal conception assumed human nature. This Logos naturally (or I should say supernaturally) floods Jesus's human soul with the "full level" of grace that we can also ASPIRE to have, if we let the Holy Spirit in us does his work. So Jesus got the "full strength" of grace from the get go. Because his human soul was not tainted by original sin, and because none of his acts are sinful (sinful acts darken human souls like tar), his human soul is able to SEE and ENJOY his "divine side" perfectly (beatific vision). As a consequence Jesus could enjoy the life filled with full intimate relationship with God the Father, testified in Scriptures a lot.

2. It is very important to assert that there is only ONE SUBJECT (the Logos) who also operates "in" his added human nature. It is God's plan that everything related to our salvation is done THROUGH Jesus's human nature. The Logos uses Jesus's human nature as Godly INSTRUMENT to elevate EVERY ASPECT of our corrupted human nature described above. This is summed up in the Patristic teaching of St. Gregory that "what has not been assumed has not been healed" or in another Patristic teaching that "like sanctifies like" (forgot the exact phrase). This is the full meaning Jesus's words: "I am the WAY, the TRUTH, and the LIFE" (John 14:6a).

3. In particular, Jesus's human soul:

   • receives prophetic revelation from the Logos, that in turn He communicated to the apostles, other disciples, and everyone else he encountered. This is how our minds receive the saving knowledge from God, which enables us to SEE farther into the ultimate end (our salvation). Other prophets can only warn and point (like John the Baptist), but unlike other prophets, Jesus (being the light of the world) can ALSO infuse us with the light of faith so we can trust God and invite Him to dwell in our soul. As a consequence we can then live righteously (or at least ENVISION to live righteously while fighting our sinful desires) by doing what Jesus commanded in his Sermon on the Mount (Matt 5-7) and in his Farewell Discourse (John 14-17).
   • shows us how to love, by voluntarily giving up his human life for us while we are still sinners, acting as a friend while we are still enemies
   • compassionates with our troubled selves who are either suffering injustice (the victim that the Samaritan helped) or being beset by the weakness of sins. By "compassionating" I mean that Jesus is willing to put Himself in our shoes to feel our pain as a friend (fellow human being) AND ALSO 1) showing us the way out (curing our blindness) and 2) pulling us out from the powerful river of sinful currents with his strong arm.
   • nourishes our soul with the bread of life through the Eucharist as the means of grace for our sanctification so our rational souls can know God and love God better

Conclusion

I hope the above has shown you an example definition of "soul" that is consistent with Scripture and Christian Viewpoint (i.e. Christian Anthropology). I also hope that it is now more clear how the Holy Spirit dwells in, and interacts with our soul, for our salvation and for our enjoyment of God. By these example definitions I propose (for clear thinking) that "soul" has to do with human nature while "spirit" has to do with divine nature given to us. Jesus being both Logos (the spring) AND human nature (the human bridge) can then be the only possible mediator of Life ("No one comes to the Father except through me", John 14:6b). Although we can only be "son/daughter" of God by adoption (while Jesus is Son by essence), by grace we can have communion with God the exact same way Jesus does in his human nature, so we too can ALREADY start enjoying a life in intimate relations with the Father (giving us peace and joy), which will culminate in the beatific vision in the age to come.