Is a monster taking damage and a monster having its toughness reduced the same thing?
Reducing toughness is very different from dealing damage.
Toughness tells you how much damage can a monster withstand. From the Reference Guide, &quot;Monster&quot; (page 8)

When a Monster has lost Health equal to or greater than its
toughness, it is defeated.

Toughness almost never changes. Several monsters (mainly epic ones) have variable toughness. Barring that, advancing the mystery is the only effect that can reduce a monster toughness.
An official response from FFG (Nikki Valens) on this matter:

A Monster’s toughness is the printed value in the top-right corner on the token’s back (surrounded by green tentacles). A Monster’s toughness is unaffected by the amount of Health the Monster has lost.
In the case of advancing a Mystery that includes an Epic Monster, the Epic Monster’s toughness is reduced by a number equal to the number of Health tokens placed on the Mystery card. This effect specifically reduces the Epic Monster’s toughness; it does not cause the Epic Monster to lose Health. In this way, advancing the Mystery can bypass any restrictions the Mystery might have for harming the Epic Monster.