For a time dependent vector field $v:\mathbb{R}^+\times \mathbb{R}^d\to\mathbb{R}^d$, and a say a (time dependent) probability density $u$, why do people call
$$ \partial_tu=\text{div }(uv), $$ the continuity equation, and
$$ \partial_t u=v\cdot \nabla u, $$ the transport equation?
Surely the terminology distinguishes between two fundamentally different dynamics can anyone broadly explain the main difference?