  If $X$ is a set with finite Assouad dimension, it is known that the Assouad
dimension of $X-X$ does not necessarily obey any non-trivial bound in terms of
the Assouad dimension of $X$. In this paper, we consider self-similar sets on
the real line and we show that if a particular weak separation condition is
satisfied, then the Assouad dimension of the set of differences is bounded
above by twice the Assouad dimension of the set itself. We then apply this
result to a particular class of asymmetric Cantor sets.
