I have to admit that I'm not sure what I think about the Antioch||Alexandria schema that seems ubiquitous. There are obvious differences, of course, and if (I'm genuinely ignorant here) the ancients understood themselves — as Antiochenes and as Alexandrians — in terms of their not being the other, then this schema appears more helpful. But if not, if Antioch did not intentionally see itself as an antidote to Alexandrian allegorism (and/or vice versa), then I can only wonder what this schema causes us to miss even as it reveals something about both schools.
But I lack the qualification to propose another solution, so, within the traditional schema, I'd like to offer a few thoughts regarding the two approaches to biblical interpretation. First, the debate between (or dichotomization of) Aristotelian empiricism and Platonic mysticism not only has a lengthy pedigree but also continues to influence contemporary biblical exegesis and theology. As an example, Eerdmans is currently putting out a series of commentaries, The Two Horizons New Testament Commentary, that seeks "to bridge the existing gap between biblical studies and systematic theology." I haven't looked through any of the four currently available volumes in this series, but the very perception of this "gap" corresponds to contemporary empirical analyses of texts, on the one hand, and a more philosophical theological enterprise. I'm not disparaging the THNTC series; indeed, this seems to me a strength in light of other series' failure to recognize the difference between these approaches (e.g., compare this commentary on Romans with this on I and II Thessalonians). I'm not, at this point anyway, disparaging either approach to biblical interpretation; but the differences between them should be recognized and respected.
With respect to the strengths and weaknesses of both approaches, I should first admit that I am whole-heartedly persuaded of the methodological superiority of an Antiochene approach to Scripture. I'm not here arguing for a literal rather than allegorical approach to exegesis so much as I am advocating an approach where texts are read within a reconstruction of their original socio-historical context rather than simply from an uncritical acceptance of our (the readers') contexts. Because allegorical interpretation generates meanings from a text that apply more directly (to say things kindly) to their readers than to their authors, this approach "de-natures" texts and treats them like contemporary rather than ancient artifacts. In other words, reading allegorically is akin to deciphering hieroglyphic inscriptions without need of the Rosetta Stone and the insights it affords: Who cares what the ancient Egyptians might have meant as they painted falcons and ankhs on the walls?
That said, I'm also wary of the reductionist potential inherent in emphatically literal interpretations of biblical texts. Some people, for example, insist so strongly on a particular, "literal" interpretation of certain texts that they fail to recognize the violence they themselves do to the texts they claim to be interpreting. In this way, ironically, I suggest that my own readings of the Genesis narrative are both more "literal" readings of the texts and yet do not necessitate specific literal events behind those texts. Not every text intends to communicate historical truth (see, e.g., Jesus' parable of the sower); reducing every biblical text to a historical statement fails to respect the other purposes texts — and language more generally — can serve. What is more, the significance a text can have for its readers does not depend on its original significance, at least insofar as this significance inheres between the text's author and original audience. In other words, "literal" interpretation does not end at a literal interpretation but can also raise questions the author never intended to raise. This last point, of course, lies at the heart of evangelical hermeneutics.
My last point, perhaps, is my most controversial. Let me first say that I am intentionally a biblical scholar and not a theologian. In recent years I have begun to appreciate the contribution theologians make, especially scholars such as Kevin Vanhoozer, John Franke, William Placher. (If my readings in theology appear narrow, that's probably because they are.) But earlier in my education I had only been exposed to theologians whose work seemed formulaic and rigid. That said, let me suggest that what we actually have in the Antioch||Alexandria dichotomy is not so much two approaches to biblical interpretation but rather two different methods of constructing Christianity. The Antiochene project (exegesis, in a broad sense) is biblical interpretation, whilst the Alexandrian project (theology, similarly broad) is not biblical interpretation but theological reflection. In the former, the text is primary and theological conclusions must be suggested by proposed interpretations of publicly available texts. In the latter, the text is a vehicle for expressing theological propositions that are not necessarily suggested by the text but are rather legitimated through them. The text, in this project, is ancillary to the allegorical meanings mapped onto the texts.
This tension between theology arising from the biblical text (Antioch), on the one hand, and theology finding expression through the biblical text (Alexandria), if I am correct in so naming this tension, has left an indelible mark on the Church (and its history of reading its sacred texts). Since Irenaeus at least — and arguably since the late-Pauline tradition — the purpose of biblical interpretation has often been to justify previously established theological conclusions rather than to refine and reform those conclusions. The naming of certain individuals and groups as "heretical" often — though not always — relies on theological convictions whose simplicity is at odds with the complexities of the texts themselves. I would suggest, tentatively, that many of the early Christological heresies fell victim to this move. It seems to me, for example, that Arius's portrayal of Jesus as the first born son of God, the instrument God created for the creation of the world, attempts (inadequately, in the end) to preserve the tension in the biblical text in the face of a (perceived?) denial of the distinction between God, the Creator of the world, and Wisdom/his Word, through whom God effected his creation. Similarly, the condemnation of emphatically Jewish expressions of Christianity as "heretical" seems, at least to me, at odds with the complexity of, for instance, Paul's arguments regarding the Law, the Jews, and God's covenant with humanity throughout Romans.
Perhaps some of these observations are too blunt to be of any historical, exegetical, or theological use. Perhaps the dichotomization of Antioch||Alexandria, as I first suggested, obscures some things even as it brings other things into view. But the tensions I'm trying to identify lie at the heart of my own faith tradition which seeks to marginalize Alexandria and to epitomize Antioch in slogans such as, "No creed but Christ; no book but the Bible" (see also my comments here). Even despite this slogan (or perhaps especially because of it), theological conflict in my tradition has often employed polemic and philosophical argument explicitly and neglected to pursue questions and answers through biblical interpretation. I suspect, in the end, that this is rooted in the oft-felt but rarely spoken recognition that sola Scriptura doesn't always (doesn't even usually, perhaps) support our own conclusions as clearly and as forcefully as we would like. In these cases, Alexandria becomes eminently useful; Antioch, not so much.
[For an Orthodox discussion of Antioch||Alexandria, see this article, though recognize that this site is remarkably partisan.]