I've been sorting a list of the monsters from 2e into 5e categories, so I've had to think about this.
The two trickiest distinctions are Beast <> Monstrosity and Aberration <> Monstrosity. An honorable mention goes to Humanoid <> Monstrosity, since monstrosities can be humanoid in shape.
I compared the Beasts and Monstrosities on D&D Beyond to see if there was a clear difference. Beasts can be intelligent, have alignments (rather than being unaligned), and even speak. They can look like you took a regular animal and stuck wings or a horn on it. They can have damage resistances and even cast (innate) spells (dancing lights for Deep Rothe).
My conclusion was that the principles for Beasts vs Monstrosities are basically this:
-Beasts almost always look either like real animals, or like something that could be a real animal on earth. Although rare, you can add a single out of place feature, such as a horn or wings. If you make the creature truly chimerical in form (more than just a minor feature on a clear single animal base), you have created a monster-osity.
-Beasts can very rarely have one minor supernatural ability. It might be a damage resistance, or some special feature. If they have more than one, or if they are more powerful (like a damage immunity), you have a monstrosity.
-Individual beasts who are magically transformed can remain beasts even if they gain features that make them more like a monstrosity. In the same manner, other creatures transformed into beasts can sometimes retain their abilities--like a mage who can still cast spells even though he's now a rabbit or something.
-Monstrosities either do not look like something that could be a real animal; or they do look similar to something that could be a real animal, but have supernatural features.
-You can probably safely follow a "rule of one" for beasts. It can have one thing out of the ordinary, like a physical feature, a supernatural ability, or intelligence (which might include language and speech). It should feel like it is a "natural" part of the world.
For Monstrosities vs Humanoids:
-Humanoids are bideps of any size from Tiny to Medium. Quadrupedal+ creatures like centaurs are monstrosities (or Fey).
-Humanoids can be anthropomorphic animals or any step in-between that and a human, but tend not to have extra appendages other than wings.
-Creatures that have populous societies are more likely to be humanoids than monstrosities.
Here are my impressions of 5e Monstrosity vs Aberration:
-Monstrosities look like a normal part of a fantasy world. Chimerical beasts like griffons or manticores, bipeds like doppelgangers, medusa or minotaurs, even oddities like carrion crawlers or mimics. Aberrations are less likely to look like a normal part of the fantasy world, and more likely to look like something out of alien sci-fi, though there are exceptions are both sides.
-Titans (in the 5e sense) are usually monstrosities also, such as krakens or astral dreadnoughts. They might be odd, but they are primordial entities of the cosmos associated with gods and such.
-Monstrosities are associated most often with the Material Plane, though some of them live on other planes also. Aberrations are not all native to the Far Realm, but most of them have some of that feel to them.
-As a general rule, while beasts should feel like a "natural" part of the world, monstrosities should feel like a normal part of the magic of the world (a treasure chest that eats you still feels like a normal magical threat), while aberrations should feel like they are bizarrely alien with no normal connection to the fantasy world.
Also, if you have something that is like a humanoid or monstrosity but you need it to be otherworldly, you can make it Fey, even if it doesn't fit the traditional Fey feel. Or if you have something you can't make humanoid by the rules, but you don't want it to be a monstrosity and you can make a case that it does have some of the traditional Fey flavor, you can make it a Fey (like the playable centaurs).
