Definition 1.2 in <A HREF="https://arxiv.org/abs/1303.1421">On the Distributional Hessian of the Distance Function</A> explains the difference: an inequality can hold in distributional sense, in barrier sense, or in viscosity sense.

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.