In the world of Harry Potter, magic manifests itself in physical beings. This is the subtle difference that Scamander is referring to.
Wizards for example are susceptible to magical ailments such as Dragon Pox, which simply doesn't exist in any Muggle medical dictionary. Likewise Muggles are much more prone to injury or death from physical harm - even as young children wizards often find themselves unknowingly 'magicking' themselves out of dangerous (but non-magical) situations. Neville for example was dropped out of a window by his great uncle, and bounced around the garden. Harry, when chased by Dudley and his bullies, somehow found himself on the roof of his school, out of harm's way. Remember also Hagrid's outrage at the very idea that Lily and James Potter could have been killed in a car crash.
So this magic running through the veins of witches and wizards gives them subtle differences in their physiology, making them more resilient to some things while more vulnerable to others.
As a side note, this helps explain why so many magical people live for so long - the kinds of accidents that might harm a Muggle just aren't as dangerous to them.