Everyone's a wizard

The difference is that everyone in B/X is a wizard in one particular way: everyone is made of tissue paper. I briefly considered using the optional rule that characters start with max hit points at 1st level and then decided not to, reasoning that I could always add it later if I didn't like the effects of low hit points but taking away the max hit points rule mid-campaign for new characters would cause a mutiny.

Apart from often having little difference in hit points and therefore durability against an enemy's 1d6 damage, the fighters are usually in the front rank and taking the hits while the mages are (usually) in the middle or back avoiding (usually) damage entirely. To top things off, a 1st-level fighter with a sword hits as often and does the same average damage as a 1st-level mage with a staff striking two-handed.

The result is that yesterday I rolled up some NPC adventurers to hire the PCs [1], and the NPC fighter had 3hp and banded plate, while one of the NPC wizards had 4hp and no armour. Over the course of the adventure, the fighter was always on the front lines while the wizard was in the middle rank, making the AC disparity less relevant. The result is that Corwyn the fighter was taken out early on and nearly died, while Miriam the wizard opportunistically brained goblins and survived without a scratch.

The dice could have easily fallen the other way, but the point is that fighters and mages are mostly on par in a B/X fight. The statistical equality of fighters and wizards at first level when everything is taken into account – including to-hit chance, hit points, weapon damage, AC, and the effects of aggressive/defensive roles in combat – was really apparent with these NPCs I was running.

Amplifying these factors is the fact that combats are so short in B/X: most PCs and enemies go down in one or two hits, making an even fight a very short thing, and an uneven fight even shorter. A wizard can wade in and smack a lingering opponent and have just as much chance of ending the fight then and there as if the fighter did the same. The small difference in a fighter's and a mage's durability only matters if the fight lasts long enough for the enemy to hit back more than time or two, and then the difference between 3hp and 6hp is still slight against a few 1d6 damage rolls.

Power spirals are to blame

Wizards in later editions, even as little later as 2e, are annoying because there is already much more of a disparity in survivability between classes.

Because of the increased disparity in later editions, a 1st-level party as a whole is more likely to charge in and take a fight's damage on the chin, dragging the tissue-paper wizard with them into danger that's really only dangerous to the wizard. The much-weaker wizard simply takes much less of a combat role for the very sensible reason that the fighters are so much better at it, doing their thing every round, round after round, and mostly managing to keep standing. A 1st-level wizard's spellcasting ability is therefore the only thing an AD&D wizard is good at, making it much more important. And yet, they still cast only a single spell.

Meanwhile, the usefulness of a 1st-level mage in B/X is fairly general since they fight very nearly as well as anyone else, plus they get to pull out their special trick once a day. No wonder the AD&D mage annoyed me so much! They're so very niche protected that they're good for nothing but their niche, and at 1st level that niche frankly sucks goats.