How does the Catholic church distinguish between the human soul and spirit?

The best example is Mary's Magnificat

And Mary said:

“My soul magnifies or glorifies the Lord and my spirit rejoices in God my Savior,...(Luke1:46-47)

Mary's humanity of living a prayerful life of humility,charity,poverty & obedience "magnifies" the image & likeness of God imprinted on every created soul.

" And God said, Let us make man in our image, after our likeness.."(Genesis1:26)

and by stating that Her "spirit" are the one praising or rejoicing and not the soul is fulfilling the Father's command;

John 4:24 King James Version (KJV) 24 "God is a Spirit: and they that worship him must worship him in spirit and in truth."

To answer your additional question on the Catholic Teaching on the body,soul and spirit.

Catholic Church teaches that the soul is the form of the body and therefore the life of the human person is composed of body & soul at the moment of conception.

365 The unity of soul and body is so profound that one has to consider the soul to be the "form" of the body:234 i.e., it is because of its spiritual soul that the body made of matter becomes a living, human body; spirit and matter, in man, are not two natures united, but rather their union forms a single nature.

The Catholic Church does not teaches that when God breathe into Adam nostrils, God breathe in a soul to Adam, as it will contradict it's teaching on CCC366 that soul is "immediately created at the moment of conception" and stated that the unity of soul and body is profound.

When God created the body of Adam it will be wise to consider that the soul was immediately created too to give the body of Adam a human formed and it will harmonized the definition stated in CCC365.

When God breathe into Adam nostrils, God breathe in Divine Life or Sanctifying Grace to give Adam the ability to commune and established friendship with God.

We can validate this understanding when Adam & Eve fall as they had suffered death of their soul by losing the sanctifying grace and friendship with God.

The Catholic Church teaches that thru the Sacraments of Baptism "sanctifying grace" will be pour in to the soul to cleanse it of original sin as a gift and thru Sacraments of Baptism and become adoptive sons of the Father. The relationship to God is restored back and the soul thru sanctifying grace can commune again to God.

In closing, going back to Adam when God created him before breathing in to his nostril the Divine Life/ Sanctiying Grace was already compose of body & soul. But a soul who had no sanctifying grace is not a Living Soul as "sanctification" is the one who gives life to the soul.

CCC1997 Grace is a participation in the life of God. It introduces us into the intimacy of Trinitarian life: by Baptism the Christian participates in the grace of Christ, the Head of his Body. As an "adopted son" he can henceforth call God "Father," in union with the only Son. He receives the life of the Spirit who breathes charity into him and who forms the Church.

CCC1999 The grace of Christ is the gratuitous gift that God makes to us of his own life, infused by the Holy Spirit into our soul to heal it of sin and to sanctify it. It is the sanctifying or deifying grace received in Baptism. It is in us the source of the work of sanctification:48

So, the one infused to our soul is the "sanctifying grace" to make us participates in the life of God and it is an aid for us to live and embraced a Trinitarian Life like Mary revealed in the Mystery of the Annunciation.

The Blessed Virgin Mary at the Annunciation, received first the Will of the Father and the Holy Spirit "overshadowed" Her and only then the Logos is formed in Her Most Pure Womb.

So, Mary is the first to embraced the Trinitarian Life, as a "beloved daughter of the Abba Father" the "beloved Mother of Jesus Christ" and lastly the "beloved Spouse of the Holy Spirit."

In parallel conclusion the Breathing of Life is synonymous to Jesus Breathing in the Holy Spirit to the Apostles after His resurrection.

"When He had said this, He breathed on them and said, “Receive the Holy Spirit."(John20:22)

So, in reality the spirit that comes from God as a gift brings life to the soul. The spirit animates the soul and the soul animates the body. And in the end when we die our soul will face the judgment of God and the spirit/sanctifying grace that God poured into us thru Sacraments of Baptism will return back to God. If we follow what Jesus teaches in the Gospel our soul will be glorified but if we follow the ways of wickedness our soul will go to hell. Fortunately, the Catholic Church teaches purgatory as only few able to enters the narrow path of perfection like the Saints & Martyrs. Since nothing defiled shall enter the Kingdom of God, we as Catholic believe that our imperfection or venial sins not mortal sins will be cleanse in purgatory and the cleansing of our soul will be aided by the prayers & supplications of all the people praying for us.

Regarding "spirit", the New Catholic Encyclopedia states:

For those of the Christian tradition, spirit is always personal and subjective, and all other manifestations of spirit can be reduced to their source in the person. Within this tradition, the radical and essential manifestation of spirit has been variously singled out as: creative activity, self-consciousness, interiority or subjectivity, intelligence, reason, knowledge of universals, love, freedom, and communication (dialogue). These are activities by which the presence of spirit may be known, and they furnish a clue to the nature of spirit in itself as a form of subsistent being.

Christian thought also recognizes three main kinds of spirit: (1) the human soul, incomplete in its mode of subsisting and extrinsically dependent on the body; (2) pure finite spirit, i.e., the angel, perfectly subsisting and independent of matter; and (3) Absolute Spirit, or God, infinite, utterly pure, and fully actual being (subsistent existence) without any limitation. Man’s primary apprehension of these forms of spirit is gained through self- knowledge. The spiritual being most proportionate to his way of knowing is his own soul, manifesting its nature through activities that are immediately present to his consciousness. His knowledge of other spiritual realities is in turn based on such knowledge (cf. St. Thomas Aquinas, C. gent. 3.46).

So, human beings, because have body, and the soul is the form of the body (e.g. here), have souls. The soul is the spirit.

Angels, it seems, have no soul because have no body. In effect, the encyclopedia states regarding angels:

The Church has defined as dogma that besides the visible world God also created a kingdom of invisible spirits, called angels, and that He created them before the creation of the world (Lateran Council IV, 1215, ch. 1, H. Denzinger Enchiridion symbolorum [Freiburg 1963] 800; repeated at Vatican Council I, 1870, ibid. 3002; cf., earlier, the Nicene Creed of 25, ibid. )

Same seems to apply to God. Being immaterial, It has spirit, but no soul.

PS: unless the dogma that the soul is the form of the body refers to human souls only, it might be possible for angels to have another form of soul. I haven't found a definitive teaching on this.