Why this split? This began as a comment on a Google Plus conversation, but I think it’s worth a blog post. For me, the split is based on two things: problem solving tools and archetypes. For archetypes, the inspiration is swords & sorcery. This, in my opinion, is uncontroversial and does not need further elaboration (other than to remark that the cleric, if taken too far away from the original Van Helsing and Solomon Kane inspirations, does not fit so well aesthetically or culturally).
Clerics are really a hybrid class in terms of problem solving, and could potentially be either fighter/mages (for the trad crusader vampire hunter that also has some magic) or thief/mages (a version less often seen, but just as thematic for zealous witch hunters or hashashin characters). However, the hybrid nature of the cleric means that it can be understood based on the other three main classes, so no more need be said about the cleric independently.
The primary problem solving qualities of the core classes are: combat/renewable resource (fighter), combat/consumable resource (magic-user), utility/renewable resource (thief), and utility/consumable resource (magic-user). Thus, the magic-user is more versatile, but resource-limited (and in most incarnations, more fragile). Obviously there is some bleed between the approaches when you consider the actual implementation (everyone can make melee attacks, fighters can still use some magic items, etc). So that’s where the split comes from in terms of OD&D game mechanics.
Edit: I should also link to Talysman’s post on classes and problem solving here.