Beasts versus monstrosities

Many tabletop games treat magic as separate and self-contained from the "normal" physics of fantasyland. I believe that settings may be enriched by removing such arbitrary distinctions, hence my introduction of magical background fields underpinning reality. Continuing that line of thought, I would tackle the magic versus mundane argument as it applies to animals.
UPDATED 10/20/2017

Where does life come from?

Those living in fantasyland are subject to fantasy physics. There are several phenomena that act on living creatures: bio-engineering, creationism, evolution, spontaneous generation.

• Creationism is when an intelligent designer, like a deity, creates something from scratch. Many fantasy worlds and their inhabitants were created this way.
• Spontaneous generation is similar to creationism, but occurs without the intervention of intelligent designers. 
• Evolution is when populations of living creatures, whether originally created or spontaneous, change traits over time due to random mutation and natural selection.
• Bio-engineering is when gods and mad wizards decide to mess with life to create chimerical fusions and all other kinds of bizarre creatures.

The classical philosophers believed that the animals on Earth were the surviving descendants of random designs that arose spontaneously. That is, all animals are technically chimerical hybrids and the ones we Earthlings are familiar with are merely those which survived to reproduce. Spontaneous generation and bio-engineering occurs all the time in fantasy land and results in the innumerable creatures of chaos that plague the land.

Where do the components of hybrids come from, then? I would invoke the concept of morphic fields. Unlike in reality, these are real in the fantasy world due to the winds of magic.

What is a beast?

According to Sage Advice, "Typically - but by no means an iron clad rule - anything that could potentially exist & operate without magic is a beast."

I will give 5th edition kudos for allowing animals that do not exist on Earth to be considered beasts rather than monstrosities. I have always found terracentric taxonomies to be nonsensical and was pleased by the creativity of settings like Avatar: The Last Airbender. For example, the tressym and crag cat are typed as beast despite having wings and magic reflection, respectively.

Furthermore, 5th edition allows beasts with high intelligence. One of the more bizarre decisions in 3.x was limiting animals to a specific intelligence bracket regardless of what real world studies had to say on the topic. Apes, dolphins and some birds may all qualify as sapient. Now it is easy to add the talking animals typical of fairy tales without bending the rules over backwards.

Yet I am not entirely sure what sets beasts apart from humanoids. For example, the great apes are considered beasts despite possessing opposing thumbs, capacity for sign language, even currency use in some experiments. Maybe it has to do with the capacity for philosophy and religion?

But I digress...

What is the difference between beasts and monstrosities?

5th edition divides animals and some other creatures into "beasts" and "monstrosities." We all know the real reason is for the purposes of game balance. If would be very unbalancing, for example, if the party druid could become a basilisk or cockatrice with petrifying gaze. However, some of the monstrosities are balanced for use as beasts and being typed as monstrosity seems to be solely for story reasons.

The story reasons for monstrosities are not consistent, including such criteria as "magical" or "unnatural." Some monstrosities are claimed to be the result of divine curses or magical tinkering, while others are given no rationale. For example, centaurs are typed monstrosity but are not traditionally considered unnatural monsters. The "magical" criterion only makes sense in settings with physics that neatly separate magic from nature (a conceit I find irrational) and the "unnatural" criterion is logically fallacious. While one could make the argument that whether something is natural or unnatural is up to the gods, this is never offered as an explanation.

In any case, the distinction between the beast and monstrosity types shall be justified with the winds of magic cosmology I briefly explained elsewhere. Beasts are tied to the green wind, the color of nature, whereas monstrosities are tied to the orange wind, the color of chaos and entropy. This provides a simple in-character explanation of the out-of-character mechanics, hopefully reducing cognitive dissonance in the minds of players.