OD&D, D&D (B/X, BECMI), and AD&D are essentially interoperable, although not the same game.
The best way to think of it is to remember B2 (Keep on the Borderlands).
B2 was written under the OD&D rules. B2 was originally packaged as a starter module for Basic (Holmes), which was a streamlining of OD&D. It was also used as a starter module for Moldvay Basic. In addition, it was the starting module for numerous AD&D campaigns.
In other words, D&D (in all the variants, from OD&D up to and including 2e) was always the same game during the TSR era, just with more or less stuff added on.