It has been a few years since I did rigorous work on Augustine, and a few more years before that when I was working on Plato, but let’s give this a whirl.

Firstly, it’s important to distinguish between the thought of Plato, and Platonic thought. It’s usual to discuss both Platonism, Middle Platonism, and Neoplatonism as different periods of philosophical thought that grew out of adaptation and expansion from Plato’s ideas.

We also need to be careful, as always, of imposing overly simplistic narratives on history. In this case, I think two narratives to be wary of are “Platonics vs Aristotelians” which seems nicely dichotomous but betrays complexity, and “Christianity just adopted Hellenic philosophy wholesale” which also fails to engage with the depth of theological wrangling that early Church Fathers do in adopting/rejecting/integrating philosophical concepts from the Hellenic tradition with a Scriptural tradition that they are always working with.

Augustine, like most thinkers of his time, is clearly acquainted with both Plato, and Platonic thinkers. I say “like most thinkers” because from about the 4th century onwards almost all significant theologians who are bishops come from backgrounds where they enjoyed traditional and extensive training in the rhetorical tradition, which would include Greek philosophy.

Personally I don’t think ‘influence’ is the best way to talk about this, but rather ‘engagement’. So for example, in City of God 8.5 he engages with Platonism, in 10.25, 27 he engages directly with Porphyry. Augustine is writing in a philosophical climate where he must talk about Platonism, to do so would be not to enter into the very public discussion.

Specifically on the role of neo-Platonism, you should track down and read Robert Course, ‘Paucis mutatis verbis: St Augustine’s Platonism’, in Augustine and His Critics: Essays in Honour of Gerlad Bonner 2000. Specifically on City of God, look for Dominic O’Meara’s Platonopolis: Platonic Political Philosophy in Late Antiquity.

I think you are seeing things that aren’t quite there in comparing the Platonic tripartite idea of the soul to the Trinity. Plato’s idea is of the Logical, ‘Spirited’, and Appetite. The formulation of the Trinity does not come out of analogy with the soul, but a very different set of theological/philosophical arguments through the 2nd to 4th century that are mostly resolved by the time of Augustine. When Augustine writes On the Trinity he takes those resolutions as a starting point for his great summa, and then pushes it further in applying his own analogy, which depends firstly upon Subject/Object/Means of Love.

However when he does come to his psychological analogy, he is using Platonic like language: mens, animus, anima, which does seem to draw from Plato. However, and this is my key point here: Augustine derives his psychological analogy after he has spent 7 books delineating a doctrine of Trinity. He doesn’t begin with a Platonic tripartite soul in order to develop an idea of the Trinity. Furthermore, the psychological reflection of the trinity in the human mind does not, itself, follow the tripartite division. Rather he talks more about triadic traces of the trinity’s image, for example: mind, the mind’s knowledge of itself, and the mind’s love of itself.

The problem with seeing Manichaeism as a primary influence is that Augustine wrote extensively against Manichaeism after his conversion to Christianity. For instance, in De natura boni contra Manichaeos (ca 399) he discusses evil as the absence or lack of good in a created good world, over against the Manichaean idea of two equal eternal principles of good and evil. Augustine consciously and vocally repudiated Manichaean ideas in his writings.

Just as a couple of other reading recommendations:

Lewis Ayres Augustine and the Trinity 2010

Matthew Levering The Theology of Augustine: An Introductory Guide to his most important works, 2013

I disagree with two things in /u/synthesisman’s comment, I should mention. Firstly, I think it’s overstating the case to say that Christanity appropriating Greek thought was “the key factor in [it’s] amazing spread”. Secondly, the background to John’s use of Logos in John chapter 1 is undoubtedly complex and disputed, I am cautious about all claims to have ‘definitely identified’ what philosophical background John is invoking or expecting his readers to understand.

This is a very broad response with sweeping generalizations on the intersection between philosophy, theology, life and practice, over a large period of time, so be warned and take what I say with a grain of salt, knowing that it's a question I'm passionate about.

I think that the biggest impact of Greek philosophy on Christian thought was the concept of timelessness or the dualistic view of things, including the view of God, goodness, authority, sin, salvation, judgement, law, pretty much everything. Formulations of God such as the omnis (omniscient, omnibenevolent, omnipotent) don't make a lot of sense within an old testament context that rarely if ever discusses attributes out of context (for example, God makes his goodness pass before Moses - rather than being a platonic ideal, God's goodness can be shown, through actions, etc). The idea of a God existing outside of time, as an ideal, has lead to a lot of ongoing christian theology such as the rejection of creation and a second coming as being significant to Christian thought (within a timeless mode, they no longer have any narrative significance), the changing views of God's authority (as something that exists separate from a relationship, which can be disembodied and passed on to a representative or church), the changing view of good and evil (as again, things that can be separated from consequences, measured, evaluated) and so on.

This is a biased answer, but my understanding is that prior to the significant impact of Greek thought, the theodicy question was approached as a narrative problem, in which sin arises in heaven, comes to earth through Adam and Eve's choice, God then works towards dealing with it, culminating with christ's revelation, and eventually Sin will be dealt with at the second coming or the judgment - i.e. the answer to the theodicy question is "God is telling a story, and it will be solved at the end". Within a more platonic worldview, where God is seen as being himself a universal, outside of time, the embodiment of all these ideals, it is no longer possible to describe a narrative solution, as any such solution is based on change, and the existence of God in time, and as such the question all of a sudden becomes important. "In this moment, which is the same as all other moments, how can Evil exist with a Good god who is Powerful?" - the use of caps is a part of the worldview.

Within a timeless perspective, law is seen as a timeless entity, whose existence can be separated from purpose and from consequences, and sin becomes similarly a tangible quantifiable concept linked to a range of universals.

Likely my phrasing has betrayed my attitude towards the approach. Nietzsche is probably the best known of the various continental philosophers who rejected a timeless mode of viewing, well, of viewing everything, claiming that in detaching reality from the narratives we used to make sense of reality, we left ourselves blind followers of blind leaders. His claim that "Christianity is Platonism for the masses" is a fairly accurate depiction of his experience with Christianity, and his books are a very good deconstruction of the whole timeless Platonistic endeavour.

This rejection of platonism has also taken place within Christianity as well - my favourite so far would be the work of Jacques Ellul, in particular his book On Freedom, Love, and Power, but in many ways, the impact of platonism permeates our culture, our relationship to government, our languages, our approaches to conflict, in particular our formulation of the concepts of what it means to be independent, what it means to be free, well, everything.

Interestingly enough, it's also a topic that I confront daily as a linguist doing language documentation, as the underlying basis of so many of the concepts and structures we use to convey ideas about the mind, about thought, about relationships, and about government, good, evil, and so on, are tied up with a platonic worldview in a way that becomes very evident when trying to document how to convey similar ideas in the various non-western languages I'm working with. A lot of concepts simply don't translate, or if they do they carry vary different connotations, because these languages do not conceive of good and evil existing separate from good and evil actions, and do not readily have a way of talking about authority as something that is not the result of an exchange of responsibilities.

I've not yet dived directly into Plato, so I cannot comment as to specifics of his thinking in Christianity. However, in my studies, it is well-attested that Plato's way of thinking about the world (following from Parmenides) has had a profound impact on both Western and Christian thought). Nietzsche and Heidegger both take this up at length in their respective work and the influence of Greek philosophy (via Plato) on theology has continued to be explored particularly since the late 1800s. I have Fernando Canale's work most helpful in its attention to the details of this influence. He contends that the most significant influence is in terms of ontology, or what he terms "primordial presuppositions." His doctoral dissertation argued that Christianity has been profoundly shaped by the adoption of a presupposition of timelessness from Greek philosophy, in opposition to the temporal presupposition of the Old Testament. (Interestingly, he contends that Aristotle also assumes timelessness). Oscar Cullmann argues in a similar vein regarding New Testament scripture.

This difference of perspective about the nature of ultimate reality (ontology) has profoundly impacted Christian theology. Based upon a timeless conception, God is figured as being external to history (time) and thus his involvement in time is only symbolic or figurative. He is not truly "with" us, because he cannot be. Humanity is figured as being composed of a material body and an immortal soul--which is the means of our "relationship" with or knowledge of God. In turn, salvation is a matter of belief in timeless ideals, rather than having any connection to a lived, historical reality. Heaven becomes an escape from the body, from materiality and time into the presence of a timeless, unchanging God. I could go on.

In contrast, if, following the Old Testament, ultimate reality is conceived as temporal, God becomes a being who is with us in history, who both responds and is responded to by humanity--Christ is the center of this involvement insofar as he lived and died as a material, human being (which need not deny his divinity if God is always in timewith us). Humanity is figured as essentially material and wholistic--containing the life giving breath of God, but having not existence apart from time. In turn, heaven is not a removal from the world, but a restoration of the world. Salvation is a historical process not dependent on agreement with particular propositions, but upon a particular way of life enabled by God's presence with us.

This difference is at the center of many current controversies in Christianity, including open theism, hell, creation, politics, etc. If belief is all that matters, if truth is about a timeless realm, rather than this material existence, great evil can be justified in the name of a "greater good." However, if ultimate reality is temporal, truth is a way of being in that reality, a way of relating to history itself--a way of being with others. Thus, our actions--the way we relate to God, others and the world--become hugely significant. This changes the way we understand the entire Christian narrative.