  We study the semigroup identities satisfied by finite rank plactic monoids.
We find a new set of semigroup identities of the plactic monoid of rank $n$ for
$n \geq 4$, which are shorter than those previously known when $n \geq 6$.
Using these semigroup identities we show that for all $n \in \mathbb{N}$, the
plactic monoid of rank $n$ satisfies a semigroup identity not satisfied by the
semigroup of $(n+1) \times (n+1)$ upper triangular tropical matrices. We then
prove that the plactic monoid of rank $n$ generates a different semigroup
variety for each rank $n$.
