  Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set
$S\subseteq V(G)$, we define the external neighbourhood of $S$ as the set
$N_e(S)$ of all vertices in $V(G)\setminus S$ having at least one neighbour in
$S$. The differential of $S$ is defined to be $\partial(S)=|N_e(S)|-|S|$. In
this paper, we introduce the study of the $2$-packing differential of a graph,
which we define as $\partial_{2p}(G)=\max\{\partial(S): S\subseteq V(G) \text{
is a }2\text{-packing}\}.$ We show that the $2$-packing differential is closely
related to several graph parameters, including the packing number, the
independent domination number, the total domination number, the perfect
differential, and the unique response Roman domination number. In particular,
we show that the theory of $2$-packing differentials is an appropriate
framework to investigate the unique response Roman domination number of a graph
without the use of functions. Among other results, we obtain a Gallai-type
theorem, which states that $\partial_{2p}(G)+\mu_{_R}(G)=n(G)$, where
$\mu_{_R}(G)$ denotes the unique response Roman domination number of $G$. As a
consequence of the study, we derive several combinatorial results on
$\mu_{_R}(G)$, and we show that the problem of finding this parameter is
NP-hard. In addition, the particular case of lexicographic product graphs is
discussed.
