It's ASIs, and level advancement. See, the problem I'm having is something that is deep and fundamental to 5e, and, well, to all D&D after 2e. The coupling of so many mechanics to ability scores, as opposed to levels.
Allow me to rip the rose-tinted glasses of your eyes for a minute...
AD&D had far more mechanics tied to ability scores than you might think. Lets take a few examples (note: I'm using 2e examples, as they are more familiar to me; 1e values might vary).
First off, each race had minimum and maximum stat requirements. Dwarves couldn't have less than an 8 Strength and an 11 Constitution, nor could they have more than a 17 Dex or Cha. Halflings required a 7 Str and Dex, 10 Con, and 6 Int, and also couldn't have an 18 in Wis and, even if they were a fighter, didn't get percentile for an 18 in Str. So right off the bat, your ability scores limited your racial choices and even how good you were at certain classes (as we'll see below).
Secondly, Classes had minimum requirements. Some were fairly easy (9 Str for fighters, 9 Wis for clerics) while others were nearly impossible (17 Cha for paladins). Again, your ability scores limited your choices here. But it wasn't just meeting the minimums; ability scores also determined how quick you could advance in a class (prime requisites) and how far you could go (expanding level limits in the DMG).
Further, each of the major class groups (warrior, rogue, priest, wizard) had abilities tied to having exceptionally high ability scores. For warriors, it was percentile Strength (for better hit/damage, you know, your job description) as well as more HP for high Con (other classes were capped a +2/HD max, no matter how high their Con score was). Intelligence controlled how easily a wizard could learn spells, how many they could know per level, and what level spell they could cast (forget 9th level spells unless you had an 18+ Int). Wisdom likewise controlled a priests high level spell access (requiring an 18+ Wis for 7th level spells) as well as your bonus spell slots. Thieves got +% to their thief skills for having high dexterity (and since skills were your bread and butter, it paid to have them max out as quick as possible). So if your classes main ability scores were sub-par (and in AD&D, we mean 15 or below) you were effectively penalized at how good you were at your class, how quickly you advanced, and what level you could reach in it*.
There is a reason I saw more rampant ability score cheating in AD&D than I ever did in 3e and on; the game damn near forces you to have 16+ in at least one, usually more, ability scores. And with no way to raise said scores beyond magic or DM fiat, lets just say I never saw a wizard with less than a 17 Intelligence...
(This also creates a few interesting corner cases, where a halfling cleric could never cast 6th or 7th level magic, as they were capped at a 17 Wis and coupled with level limits with boosts; never exceeded 10th level anyway. Barring powerful magic [DM fiat] of course)
Oh, and there's the little thing we call dual-classing ability limits; 17 in original class primes, 15 in the new class. And you through 5e's multi-classing limits was bad!
The difference, I guess, was that ability scores were more covert in how they controlled you, and if you never advanced beyond 10th level (as most AD&D games of that era did) or you had a forgiving DM who bent the rules (as most AD&D games of that era did) then it wasn't so bad. But if you played beyond level 13 (god bless you) the game punished you for not rolling well. Whereas 5e gives you a lower starting point (most PCs begin with a 15-17 for point buy, plus ASI) AND removes many of the above restrictions (want to play a dwarf with a 6 Str? Allowed! A paladin with a 14 Cha? Go for it, A wizard with a 15 Int? Still get your 9th level spells).
Its easy to think how ability scores seemed less important in AD&D (which, as pointed out, wasn't the case.) Basic D&D, however, never had many of the above restrictions and bonuses started much lower, so it is easier to say they didn't matter as much (though the game still have class minimums and prime requisites, ability scores didn't affect your class function as overtly as they did in AD&D.) I think though that 5e has done a decent job of balacing ability scores meaning something AND not penalizing your for having less the maximum in your important scores.
YMMV and all that.