Why does coherence begin to matter at the tricategorical level?

It is well known that every weak $2$-category is equivalent to a strict $2$-category, with the equivalence essentially given by (unless I'm mistaken) the $2$-Yoneda embedding for an arbitrary $2$-category into its strict $2$-category of $2$-presheaves into $\mathfrak{Cat}$ -- this means we can effectively ignore the coherence diagrams for associativity and unitarity in $2$-categories and work with strict ones where composition is associative on the nose and units vanish under composition.

This is not so for $3$-categories; Gordon, Power and Street proved that all weak $3$-categories are equivalent to Gray categories, where associativity and unitarity still hold on the nose, but we have an additional coherence notion in play called 'interchange' for $2$-morphisms which introduces fundamental differences between $3$-categories with strict interchange and weak interchange laws.

Intuitively speaking, why does this type of coherence 'matter' more than the associative and unital coherence that appears at the $2$-categorical level?

The notions of '$n$-equivalence' become more varied as we move into higher $n$ as I understand it, so perhaps the 'correct' notion of $3$-equivalence is sensitive to interchange in some way that $2$-equivalence isn't sensitive to associativity for $1$-cells?

I am currently working through the linked Gordon-Power-Street paper on coherence for tricategories and suspect the answer is buried in their proof of the tricategorical coherence theorem, but I thought someone here might already be familiar with it and able to help a newcomer -- any assistance is appreciated.