Consider a druid's Wild Shape
and other ways to access creature abilities
As you cite, the Monster Manual explicitly explains that Beasts are "natural" while monstrosities are "not ordinary, not natural". Oftentimes monstrosities are fantastical beasts drawn from real-world mythology, so there is some coding there that has to do with theme, setting, and flavor. Depending on the nature of a DM's world, there might be more or less room for monstrosities, either in the world as a whole or in certain areas - their lack can make an area seem more naturalistic, their presence can heighten the sense of fantasy.
While there appear to be many reasons for separating the two (and, for that matter, for specifying a difference between monstrosities and aberrations), one of the most practical / game design reasons has to be that of character access to special abilities. Simply put, monstrosities, as fantastical creatures, often times have special abilities that beasts do not, special abilities that might prove unbalancing if they were reliably available to characters, especially low-level characters. As the Monster Manual says
Certain spells, magic items, class features, and other effects in the game interact in special ways with creatures of a particular type.
A fundamental game design feature is that it is fairly easy for characters to take the form of beasts (as well as to communicate with them or summon them, see below). Groody the Hobgoblin, in their answer, mentions Polymorph, a 4th level spell that is typically available to a 7th level caster. Far sooner than this, though, druid characters have access to Wild Shape. Beasts might have more naturalistic abilities like knocking opponents prone or pack tactics, or they might have swimming or flying speeds (which Wild Shape carefully limits to higher levels), but they don't generally have features out of balance with other options available to a second-level PC.
Without the distinction between Beasts and Monstrosities, however, or were Wild Shape to permit taking the form of a monstrosity, consider what abilities would be accessible to a second-level Circle of the Moon Druid. Often time these abilities mimic what might be accomplished with level-appropriate spells, but could be used multiple times in an encounter without having to spend spell slots:
As a CR1/2 Rust Monster (and available to all druids by level 4)
Rust Metal. Any nonmagical weapon made of metal that hits the rust monster corrodes.
Antennae. The rust monster corrodes a nonmagical ferrous metal object it can see within 5 feet of it.
As a CR1 Death Dog:
If the target is a creature, it must succeed on a DC 12 Constitution saving throw against disease or become poisoned until the disease is cured. Every 24 hours that elapse, the creature must repeat the saving throw, reducing its hit point maximum by 5 (1d10) on a failure.
At 6th level, a Circle of the Moon Druid can wild shape into CR2 beasts. If this included monstrosities, they could add the following abilities,
As a CR2 Carrion Crawler:
the target must succeed on a DC 13 Constitution saving throw or be poisoned for 1 minute. Until this poison ends, the target is paralyzed.
As a CR2 Ettercap:
The creature is restrained by webbing.
As a CR2 Mimic:
Shapechanger. The mimic can use its action to polymorph into an object or back into its true, amorphous form...
Adhesive (Object Form Only). A Huge or smaller creature adhered to the mimic is also grappled by it (escape DC 13). Ability checks made to escape this grapple have disadvantage.
False Appearance (Object Form Only). While the mimic remains motionless, it is indistinguishable from an ordinary object.
Grappler. The mimic has advantage on attack rolls against any creature grappled by it.
At seventh level, any druid (as well as bards, sorcerers, and wizards) gains access to the polymorph spell, allowing creatures with a flying speed, which the Circle of the Moon druids can then wild shape into at level 8. Considering just monstrosities that then would be available for wild shape, we can add the
CR1/2 Cockatrice
the target must succeed on a DC 11 Constitution saving throw against being magically petrified. On a failed save, the creature begins to turn to stone and is restrained.
and the CR1 Harpy
Every humanoid and giant within 300 feet of the harpy that can hear the song must succeed on a DC 11 Wisdom saving throw or be charmed until the song ends.
Of course, by the time a druid caster gets polymorph, they can start making themselves or party-members into creatures of far higher CR than 2, meaning monstrosities with far more potent special abilities would be available if permitted.
I have limted myself in these examples to just those monsters within the Basic Rules and Monster Manual, that is, those that were official at the time the designers made the distinction between beasts and monstrosities. Since then, many others have been added, such as the young basilisk from Out of the Abyss. Since the distinction already exists, however, new beasts and monstrosities can be added to the game without affecting the balance of the classes so long as the designers keep in mind the principle that powerful special abilities are better kept to monstrosities.
When characters are not actually becoming beasts, they could be speaking with them to gain their favor (Speak with Animals, available at 1st level to Bards, Druids, and Rangers, as well as to certain Clerics and Paladins), or even summoning them as combat allies (Conjure Animals, available at 5th level to druids and later to rangers). Consider a 5th level druid conjuring four cockatrices with a single spell, getting four chances a round to turn opponents to stone. These spells are reasonably balanced when applying them to just beasts; if they could equally affect monstrosities, that would greatly change the power of the druid class with relation to the others by making many more special abilities either available or avoidable.
