The Catholic Encyclopedia entry on Apollinarianism provides an excellent background why emphasizing the "taking of the Manhood into God" was necessary. At the time that the anathema against the heresy was declared in 381, the doctrine of Hypostatical Union was not yet fully developed, so there were a lot of people who believed that Christ's soul was not 100% human, but replaced by the Word.
To answer your question, the creed addressed the ambiguity inherent in the manner of incarnation.
John 1:14 on its own does not seem to sufficiently guarantee that Jesus's soul was fully human. Reading Pope Damasus's anathema in full makes it clear that the Church needed to emphasize that Christ came in the flesh to also "save the rational and intellective soul of man" which probably what the language "taking of the Manhood into God" refer to.
An extensive discussion can be found in The History and Theology of The Three Creeds Vol II, 1854 by Rev. William Wigan Harvey, M.A, chapter titled "The History of the Athanasian Creed" (starting on page 541).
The chapter begun by discussing the original language, manuscript copies, the liturgical usage in various churches in late antiquity, and the dating which scholars tried to connect with the dating of when various heresies flourished as indicated by some writings of various bishops and church fathers. Then the chapter made the case of why portions of the creed were tied to the condemnation of the Apollinarian heresy and the confutation of the error of Eutychianism
(page 554-555) The next particular instanced by the Council of Chalcedon is another, in which Apollinaris and Eutyches alike erred, in imagining a mixture or confusion of the two natures in Christ. It is evident that this notion is condemned in the words of the Creed, 'Unus autem, non conversione Divinitatis in carnem, sed assumptione Humanitatis in Deum, unus omnino, non confusione Substantiae, sed unitate Personae.' Both heretics imagined a confusion of substance; the earlier, by declaring the body of Christ to be uncreate and essentially divine, even before the Theophania; the latter by asserting that, after the birth of Jesus, the substance of the Divine Word was, in such a manner, made one with the human body of Christ, as to form of two but one substance. The later was manifestly a modification of the earlier heresy. ...
Then in the next chapter titled "The Theology of the Athanasian Creed", starting on page 585, Rev. Harvey provided the theology of every line of the creed. This is the beginning of the explanation for lines 33 (Unus autem, non conversione Divinitatis in carnem, sed assumptione Humanitatis in Deum) and 34 (Unus omnino, non confusione Substantiae, sed unitate Personae). Rev. Harvey pointed out how the two lines should be read together, so I believe the intention of line 33 makes more sense now (see the bold portion below).
(pages 661-662) We are confirmed again in the opinion that the Creed was written prior to Nestorian times, by the tenour of this article; which first refutes the Apollinarian συναλοιφὴ of two natures, and then subjoins the contrary Catholic tenet, in words that even Nestorius would hardly disclaim; 'One; not by conversion of the Godhead into flesh; but by the taking of the Manhood into God.' The impersonation, so to speak, of the Son of Man, was effected by this assumption of the flesh. The creating of the flesh, and the taking it into God, was but one act. The Son of Man, before non-existent, now first had a being, the Word, or eternally subsisting Son of God, having taken upon him the flesh; but his personal subsistence was the same; He was the Son of God and the Son of Man, but the person was one, and that person was Christ. There is a parallelism also of phrase between this verse and the next, that clearly marks this. The words, 'Unus autem non conversione Divinitatis in carnem,' correspond with the term, 'Unus omnino non confusione substantiae,' the two are in fact identical in meaning; we may look therefore for a correlative identity in the subordinate members; and the statement in the one, 'Sed assumptione Humanitatis in Deum,' will mark the same reality with the words of the other, 'Sed unitate Personae.' Unity of Person therefore in Christ, both God and Man, has subsisted from the moment that the first rudimental element of humanity was created and 'taken into God.' Now this truth is very precisely and clearly worked out by Tertullian, in words that run too closely to the purpose to be abridged: 'But we find him expressly declared to be both God and Man; according to the words of the Psalm: "Quoniam Deus Homo natus est in illa, et aedificavit eam voluntate Patris." [footnote Ps lxxxvi. 4. ...] Surely in every respect, the Son of God and the Son of Man; since [he was] God and Man, and without doubt, according to either substance, distinct in its own property; for neither is the Word any thing else than God, nor is the flesh any thing else than man. So the Apostle teaches us with regard to the twofold substance; He, "who was made of the seed of David," must be Man, and the Son of Man. He who hath been demonstrated already to be the Son of God according to the Spirit, must be God, and the Word, the Son of God. ... [the paragraph continues the argument, too long to quote here] [bold added by me]
The book has been digitized by Google and available to be downloaded in PDF format.