While it is extremely easy to import things from one of these to the other, I have always seen them as separate games.
As a huge fan of Basic D&D (or Classic D&D, as I prefer calling it), I have always felt that this game has a unique identity and feel of gameplay. This became especially clear to me in the BECMI iteration with its unique Larry Elmore cover illustrations and Elmore and Easley interior illustrations and its iconic division of the rules into boxed sets for each tier of game play.
While Basic D&D is often associated with low level play, I always loved that the boxed sets promised the chance to advance to extremely epic adventures, even if only a few of my campaigns reached that far. D&D was never just about the game actually played, but also the dreams and aspirations of players for their characters. Nothing felt more alluring than the cover of the Master Set where the "player insert character" is shown riding a dragon high in the skies above the world or the cover of the Immortal Set where the hero is seen ascend beyond his mortal coil.
Basic D&D was often seen as a simpler ruleset, but the brilliant design means it is very simple at lower levels, but adds complexity at higher levels. This makes it one of the few editions that is actually highly playable at all tiers of play (4E and 5E may also rival this, but I have never reached those levels in these editions). Compared to late era AD&D 2nd Ed with its many splatbooks and additional expansions, Basic D&D felt highly manageable as most of the things you would ever need were contained in the excellent D&D Rules Cyclopedia.
Basic D&D was to me personally, also strongly tied to my favorite setting, the World of Mystara (Known World) which encompassed both the dangers of low level dungeon exploration and the sense of wonder of a truly magical world.
Don't get me wrong, I have also had tons of fun with AD&D which I played for decades (mostly 2nd ed). I have also enjoyed and played most later editions of D&D, but Basic D&D/Classic D&D will always hold a special place in my heart.
-Havard