With Magic initiate and Sentinel the number of attacks and damage grows.
Doesn't it pay off?
The Monk definitely benefits from anything that increases damage per attack, like say taking Magic Initiate for Hunter's Mark. Monk also multiclasses a bit easier than Barbarian, IME, with a wider range of classes. Even builds that would otherwise be dependent on multiple attributes, the Monk can benefit greatly from, like Paladin or Warlock (Hexblade).
A Hex-Monk can make 3-4 attacks per round at 1d6+Dex+Prof+1d6 at level 5, possibly also adding Cha mod if you take enough warlock levels, and has a bunch of stuff other than damage they can do when damage isn't the most important factor in a scene, whereas the Barbarian is mostly
just damage and lifting heavy things.
Going back to single-class monks, again, you also have to take into account moblity, because both classes aren't traditionally known for ranged attacks. But, the Monk can be effective at long range, where the barbarian is gonna have a fairly middling, if not low, Dex. Meanwhile, a purely Melee Monk is going to have a much easier time
getting to the enemy than the barbarian, who in some fights will be consigned to the boring fate of clean up crew, killing minions until or unless the party can get the enemy to where the barbarian can get to them.
OTOH, the barbarian has access to greatswords and polearms, and thus Great Weapon Master and Polearm Master. The Monk only has access to something similar if they are a ranged Kensei, and take Sharpshooter.