The thing I miss most from BECMI is the progression in the types of adventures you do. AD&D 1-4e all just handled "You go out, adventure, and earn levels, then you do it again." BECMI D&D, by default, assumes more of a life of growth. Start as young scrabbling adventurer, then gain fame and notariety, eventually can settle down with stronghold, land, rulership. As you gain levels, your scope and vision move from the village in which you were born, to your nation, to your world. Culminates in quest for immortality if you wish. You can just adventure for loot forever, but this gives more guidance on what to do something with it. (And don't worry, even at high levels, you'll still need to adventure, but at this point it's not "I want more money" but "I want to expand my kingdom" or something like that.)
Looking at the adventures drives this home for me. At higher levels, you're not just going into a new dungeon and bashing bigger monsters. You're running dominions, leading armies, dragging your fleets between the planes. I haven't seen that in the core rules in a long time. There are 3rd party rules which help support them... But as with any 3rd party book, that means it's a tiny segment of the gamers.
Exactly. The levels in the game have gone up, but the scope has been more limited.