To answer this question we must recall two statements by Jesus in his High Priestly Prayer:
"I revealed your Name to the men whom You have given Me out of the
world." (Jn 17:6)
"And I made known to them your Name, and will make it known, so
that the Love with which You loved Me may be in them, and I in them."
(Jn 17:26)
Now, in what sense did Jesus perform a fuller revelation of the divine Name already revealed to Moses in the burning bush?
To answer that, let's recall that the name YHWH, if vocalized YaHWeH, is hifil stem, third person, singular, imperfect form, of the verb "hwh", an earlier variant of the root "hyh", "to be", meaning "He causes to be". Thus, while the Name of Ex 3:14, "I Am", denotes God as He Is in Himself - Absolute, Subsistent Being -, the Name in Ex 3:15, which was the one used by the Israelites, denotes God as viewed by creatures: He who causes them to be.
The problem here is that causing creatures to be is not something that God does by necessity of nature, but something that He freely decided to do. For Catholics, this notion was defined in the strongest terms by the Ecumenical Council Vatican I, Constitution "Dei Filius", Chapter 1 "Of God, the Creator of all Things", canon 5:
If any one [...] shall say that God created, not by his will, free
from all necessity, but by a necessity equal to the necessity whereby
He loves Himself [...]: let him be anathema.
http://www.ccel.org/ccel/schaff/creeds2.v.ii.i.html
Since causing creatures to be is not something that God does by necessity of nature, but something that He freely decided to do, it seems that the name YaHWeH does not express the essence of God in Himself, but only his role from the viewpoint of creatures consequent to his free decision to create them. Since we exist precisely because of that free decision, YaHWeH expresses Who God is for us: He who causes us to be, whereas the essence of God in Himself would be expressed only by the Name in the first person revealed in Ex 3:14: Ehyeh, "I Am", or by its correspondent form when pronounced by a creature in the third person, which is the Name revealed in Ex 3:15 vocalized in qal stem: YiHWeH, "He Is", or more completely, "He Was, Is and Will Be".
Now, given that available evidence, e.g. Hebrew teophoric names, points to ancient Israelites pronouncing the Tetragrammaton as YaHWeH, is there a sense in which YaHWeH expresses the essence of God in Himself? Indeed there is, and we came to know it precisely from Jesus' revelation that God eternally begets a consubstantial Son. Now, this eternal generation is not by contingent free will but by nature:
"The Father begot the Son, not by will, nor by necessity, but by
nature."
From the formula called the "Faith of Damasus" [Of uncertain author
from Gaul about 500], Dz 15.
"Also, this Son is Son of God by nature, not by adoption, Whom we must
believe God the Father begot neither by will nor by necessity; for,
neither does any necessity happen [al. capit, 'take hold'] in God, nor
does will precede wisdom."
From the Creed of Faith of the Council of Toledo XI, 675, Dz 276.
http://patristica.net/denzinger/
Thus, there is a sense in which YaHWeH, "He causes to be", expresses the essence of God in Himself, and it is by changing the verb "to be" in the contingent sense by "to Be" in the subsistent sense: "He causes to Be", by nature, his consubstantial Son, and causes Him to Be the same reality as He Is, the only Subsistent Being, begetting Him in eternity, not creating Him.
Jesús, by revealing Himself as "I Am" (Jn 8:24,28,58; Jn 13:19), one with the Father (Jn 10:30), simultaneously reveals that God is not only "He causes to be" contingently ad extra, but also, and above all, "He causes to Be" (capitalized "Be" to denote Subsistent Being) esentially ad intra. The essence of God is not only "Being" but also "causing to Be", begetting his consubstantial Son. Thus, "Father" from the viewpoint of Jesus, and "God the Father" from our viewpoint, is the full sense of the name YaHWeH.
