  This paper derives new bounds on the difference of the entropies of two
discrete random variables in terms of the local and total variation distances
between their probability mass functions. The derivation of the bounds relies
on maximal coupling, and they apply to discrete random variables which are
defined over finite or countably infinite alphabets. Loosened versions of these
bounds are demonstrated to reproduce some previously reported results. The use
of the new bounds is exemplified for the Poisson approximation, where bounds on
the local and total variation distances follow from Stein's method.
