Definition 1.2 in On the Distributional Hessian of the Distance Function explains the difference: an inequality can hold in distributional sense, in barrier sense, or in viscosity sense.
Given a Riemannian manifold $X$, a point $x_0\in X$ and a continuous function $\phi:X\rightarrow \mathbb{R}$, we say that $\Delta\phi\leq 0$ at $x_0$ in barrier sense if for any $\epsilon>0$, there is a $\mathcal{C}^2$ function $\psi_{x_0,\epsilon}$ on a neighborhood of $x_0$ such that $\psi_{x_0,\epsilon}(x_0)=\phi(x_0)$, $\Delta\psi_{x_0,\epsilon}<\epsilon$, and $\psi_{x_0,\epsilon}\geq\phi$.
A functional inequality that holds in the barrier sense need not hold as a classic inequality, which is why the words "when necessary" are used in the quote of the OP, to weaken the inequality statement and make it apply when a classic inequality fails.