A bit harsh, but I've always kind of thought that AD&D was the version EGG came up with after a few years of development and rules-morphing, while B/X/BECMI was created for those that wanted something that remained closer to OD&D.
As I wrote above, it really depends on how you define OD&D. AD&D is largely just codified OD&D- when you include the supplements.
B/X, for legal reasons, was largely based on the LBBs (the pre-supplement OD&D books) with further simplifications and changes, and ignored the intervening years, including the OD&D supplements.
This is all pretty uncontroversial - especially when you examine OD&D (not to mention the whole Arneson lawsuit!). And given that I have already addressed these points in other posts, I'll leave it at that.