What is the Ontological Difference between the Catholic and the Orthodox View of God?
There are two well-known differences between Orthodox belief and Catholic belief, this are the doctrine on original sin and view on the Trinity with regards to procession of the Holy Spirit.
With Theosis I think there's no much different in understanding and teaching.The finer details is just the way the Church Father approaches it but the end view of both Churches, Eastern and Catholic is the person body & soul becomes divinized or achieve Theosis and become like gods. Both mentioned the importance of communion or living a Trinitarian Life.
The best example of Theosis after Jesus is Mary although St.Paul's life is much closer to us as he started from a sinful body while Jesus & Mary were both sinless.
Mary, because She is the perfect disciple of Christ, and Mary offers a short-cut to Theosis in a Godly way thru "humility & obedience" unlike the serpent who offered Eve a short cut thru "pride & disobedience".
The shortest way to Theosis is thru "Total Consecration"(St.Montfort), meaning our life is fully set aside and fully offered only for the fulfillment of God's Will. As, Father of MIC said the shortest way to heaven is "from doing our own will to doing God's Will". The "union of wills" is the simple and perfect divination.
Meaning like Mary, we must first pronounced our FIAT, allow or embraced the working of the Holy Spirit to enflesh the WORD in our lives.
To achieve Theosis we must have a Trinitarian Life and it's simply means we must follow the WILL of the Father, the WISDOM of the Holy Spirit dwelling and empowering us, and the WORD made flesh in us.Our full being in full communion with the Most Holy Trinity. (Blessed Ivan Merz)
From sinful body wash away by the blood of Christ thru Baptism and empowered by Eucharistic Life to create a "new body" thru transformation into a transfigured body a foretaste of a "resurrected body" (CCC999 & CCC1000).
So, I think the focus would be more on the ontological view of God it's relationship and hierarchy.
"Ontology is the philosophical study of being. More broadly, it studies concepts that directly relate to being, in particular becoming, existence, reality, as well as the basic categories of being and their relations. Traditionally listed as a part of the major branch of philosophy known as metaphysics, ontology often deals with questions concerning what entities exist or may be said to exist and how such entities may be grouped, related within a hierarchy, and subdivided according to similarities and differences.
Since ontological understanding is considered the major misunderstanding in Church History between Orthodox and Catholic Church. This matter has been discussed here in CSE and answered well by AthanasiusofAlex below is an excerpt of his answer;
How the Greek and Latin Fathers understood “procession”
It is important to keep in mind that the Greek and Latin Fathers understand the concept of procession differently. This difference is at the root of the misunderstanding that resulted between the Eastern Orthodox and Catholic Churches since the Great Schism.
The Greek Fathers, starting with St. Basil the Great, all agree that the Holy Spirit has its ultimate origin (Greek: ekporeúetai) in the Father. The etymology of that term is important: it is a compound of ek, which means “out of;” and poreúomai, which means “to go” or “to come.” The compound means, therefore, “to come from” or “to come out of,” in such as way as to stress the original source.
The Latin Fathers, on the other hand, developed a Trinitiarian theology that focused on the communication of the Divine Essence from the Father to the Son, and through Him to the Spirit. That communication they called processio (using Tertullian’s terminology). Again, the formation of the term is important: in Latin, the preposition pro means “before” or “in front of;” and the verb cedomeans “to go.” Procedo (from which processio is derived), therefore, means “to go forward.” Unlike ekporeúomai, the Latin term does not so clearly stress the original source.
Hence, we can say, when the Eastern Fathers consider of the Holy Spirit, they ask “Where does He come from originally or ultimately (ekporeúetai)?” The answer is “the Father.”
The Western Fathers, however, ask “Whom is the Holy Spirit in front of (i.e, from Whom does he proceed)?” The answer is “both Father and Son.”
How do Eastern Christians explain Jesus "sending" the Holy Spirit in John 15:26?
The official Church reference to this explanation is link here;
THE GREEK AND LATIN TRADITIONS REGARDING THE PROCESSION OF THE HOLY SPIRIT
Pontificial Council for Promoting Christian Unity
Now this misunderstanding or different on the approach stems from the Gospel of John15:26
"But when the Helper comes, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father, he will bear witness about me."
This can be understood as Jesus in his humanity will petition the Father to send the Holy Spirit on our behalf, just like when Jesus was sent by the Father to become the incarnate Word.
The sending of Jesus can be paralled to His messages to the Apostles;
Again Jesus said to them,“Peace be with you. As the Father has sent Me, so also I am sending you.” (John20:21)
Even the other passages as explain in the link had understanding that it would seems that the Holy Spirit is subordinate and doesn't have a Sovereign Authority,
The Promise of the Holy Spirit
"I still have much to tell you, but you cannot yet bear to hear it.(Just like Peter said to believers I can only give you milk, so Jesus cannot yet give solid food to the Apostles)
However, when the Spirit of truth comes, He will guide you into all truth. For He will not speak on His own, but He will speak what He hears, and He will declare to you what is to come. He will glorify Me by taking from what is Mine and disclosing it to you. (John16:12-14)
From this passage, the Holy Spirit is described somehow in a lesser sovereignty than Jesus as He cannot speak on his own.
The Holy Spirit in His title as the "Spirit of Truth" will testify, meaning He will INDWELT in all believers to testify to the Truth, as Jesus promised the Apostles that they will all testifies.
But the striking passages that Jesus gave is, when He said that there is one that will testify about Him that is greater than John the Baptist.
Now, among the Apostles none is greater than John the Baptist?, but there's one who knows Jesus from "Womb to Tomb", who is "living the Trinitarian Faith" by being the perfect disciple of Christ and following the Will of the Father and was Overshadowed by the Holy Spirit.
The Theotokos is greater than John the Baptist and can fully & perfectly Indwelt by the Holy Spirit "to testify" to the Fullness of who Jesus is.And this happened as scriptures attested in the UPPER ROOM.
In view of the above difficulty in fully understanding action of the Holy Spirit, there emerges a Doctor of the Church a great giant Marian devotee in the person of St.Alphonsus Liguori, who offered a critical insight but not fully supported by the Theologians.His controversial word and direct understanding of John14:16 unlock the mystery of the identity of the "another advocate".
"And I will ask the Father, and he will give you "another advocate" to help you and be with you forever—(John14:16)
"In His eagerness to show you mercy, God has given His son as your Advocate. And to make your confidence even stronger, He has given you "another advocate", who obtains through her prayers whatever She asks.Go to Mary, and you will see salvation." (St.Alphonsus Liguori, Doctor of the Church)
Another great Marian devotee a giant Saint in the person of St.Louis De Montfort(candidate too as Doctor of the Church) teaches a very strong union between the Holy Spirit and Mary when it comes to dispensation of God graces won by Jesus Christ, is only given and channel thru Mary as the Mother of the Church and rightly called by Church Father as "Mediatrix of all graces".
But the in 20th Century there emerged a great Marian devotee a great Theologian, whose teaching if given consideration by the Church to include in His understanding of Mariology will pave the way in the perfect and full knowledge of interpreting why the Holy Spirit is somehow subordinate in action and was personified as the one who will testifies and will give glory of who Jesus is, something that up to this age had not fully understood.
I'm talking about St.Maximillian Kolbe theory on "quasi-incarnation". This doctrine will fully enlightened and harmonized the "ontological view of both the Eastern Orthodox and Catholic Church" as it will affirmed the Sovereignty of the Three Persons as a Triune God co-substantial and both originated from the Father.
The confusions on the passages on John14:16, John15:26 and John16:12-14 will all be understood as the Holy Spirit fully and perfectly acting in the Blessed Virgin Mary.
This can be seen in the Light of Christ Annointing as Messiah, the "Anointing of Jesus is the Holy Spirit himself" as Catechism teaches the faithful an official doctrine of the Church.
The Anointing of Mary as the Mother of the Church also comes from the Holy Spirit by virtue of the mystical union as describe by St.Maximillian Kolbe in his writings.
St. Maximilian says that the Blessed Virgin Mary “inserted into the love of the Most Holy Trinity becomes, from the very first moment of her existence, always, forever, the Complement of the Most Holy Trinity.” We may paraphrase the thoughts of St. Maximilian Kolbe on the spousal relationship between the Holy Spirit, and the Blessed Virgin Mary, in the words of Fr. Peter Damian Fehlner
In virtue of this spousal union formally denoted by the title, Complement, Mary is able to enter, as no other, into the order of the hypostatic union, her soul being wholly divinized, because by the grace of the Immaculate Conception, it has been ‘transubstantiated’ into the Holy Spirit.
P. Fehlner, F.I., St. Maximilian Ma. Kolbe, Martyr of Charity – Pneumatologist (New Bedford, 2004)
The article "Vertex of Love" is a great insight to understand the concept of quasi-incarnation. link;
Also please see my full answer on the "quasi-incarnation".
Is "quasi-incarnation" been attributed to mystical union of the Holy Spirit and Mary?
In closing, the ontological view of God if the Church by the Divine Providence allows the doctrine of St.Maximmilian Kolbe to be considered a dogma will harmonized the teaching of both Eastern Orthodox and Catholic Church plus it would be a enlightening doctrine for Protestant and Christian denominations to understand the God given role to Mary.
Mary was the Theotokos in the Redemptive mission of Christ.
By giving an "immaculate body" to Jesus to DWELT AMONG US.
Mary is the Mother of the Church in the Salvific mission of the Holy Spirit.
By providing Her Sorrowful & Immaculate Heart to the Holy Spirit pierced by Seven Swords to house the Sevenfold Gifts of the Holy Spirit, this time to INDWELT AMONG US.
God the Father anointed the humanity of Jesus Christ by the Holy Spirit thereby effecting the Full Trinitarian Life to Jesus humanity doing the Will of the Father.
God the Father anointed Mary "full of grace" in Her first FIAT to become the Theotokos and anointed again in Her Second FIAT with the Sevenfold gifts of the Holy Spirit to fulfill Her mission as the Mother of Church.
St.Maximillian Kolbe is a future Doctor of the Church in his doctrine of "quasi-incarnation".
Mary is truly the Co-Redemptrix,Advocate and Mediatrix of all Graces.