Well to be pedenatic and specific Basic D&D has several editions and BECMI is the 3rd of them.
The Basic line consists of:
Holmes Basic: (1977): A revision of OD&D/The White Box/The LBBs that’s got some interesting and unique feature. Usually called Holmes Basic, Rarely called Blue Book.
Moldvay Basic (1981): Many consider it the definative Basic D&D, fully incorporates the Greyhawk modifications from OD&D that also define AD&D. Followed by Expert for levels 4 -14 or so by Cook. Called B/X and occassionally the Red and or Blue Book. Arguably the lingua franca of the G+ era OSR. Most retroclones start here.
Mentzer Basic (1983): Another updated Basic in a box and followed by Expert, Companion, Master and Immortal boxed sets (Red, Blue, Green, Black and Gold) respectively. High TSR, and notably a shift towards a more “Trad” style of play using largely the same mechanics as B/X but with different emphasis. Called BECMI singularly and collectively.
Denning Basic Set (1991) mostly Basic from BECMI, presented as part of a big box set with lots of effort yo make it easy to learn. Levels 1 - 5 not 1 - 3 being the biggest change. Called the Black Box, when its remembered atcall, but remembered quite fondly by some.
The Classic Dungeons & Dragons Game (1994). The Black Book efited into a book format. A forgotten runt of the liter. TSR was struggling at this point a bit.
Both the 1992 and 1994 edition were hampered by the 1991 Rule Cylopedia which bundled BECM into a single volume (Immortals being its own, rather different system). With WotC’s acquisition of TSR in 1997, the Basic line died and 2000’s 3rd edition marks the end of splitting D&D into Advanced & Basic.
The distinctions are somewhat interesting as there are some rule minutiae that changes between them, and eeven big rules changes at first. Mostly though the change is in the way the gamevis oresented, becoming increasingly narrative driven and combat focused and setting aside exploration and puzzle solving as a goal along with almost any moral ambiguity.
This has been a grognard rambling digression brought to you by Gus L.