  This paper makes the first bridge between the classical
differential/boomerang uniformity and the newly introduced $c$-differential
uniformity. We show that the boomerang uniformity of an odd APN function is
given by the maximum of the entries (except for the first row/column) of the
function's $(-1)$-Difference Distribution Table. In fact, the boomerang
uniformity of an odd permutation APN function equals its $(-1)$-differential
uniformity. We next apply this result to easily compute the boomerang
uniformity of several odd APN functions. In the second part we give two classes
of differentially low-uniform functions obtained by modifying the inverse
function. The first class of permutations (CCZ-inequivalent to the inverse)
over a finite field $\mathbb{F}_{p^n}$ ($p$, an odd prime) is obtained from the
composition of the inverse function with an order-$3$ cycle permutation, with
differential uniformity $3$ if $p=3$ and $n$ is odd; $5$ if $p=13$ and $n$ is
even; and $4$ otherwise. The second class is a family of binomials and we show
that their differential uniformity equals~$4$. We next complete the open case
of $p=3$ in the investigation started by G\" olo\u glu and McGuire (2014), for
$p\geq 5$, and continued by K\"olsch (2021), for $p=2$, $n\geq 5$, on the
characterization of $L_1(X^{p^n-2})+L_2(X)$ (with linearized $L_1,L_2$) being a
permutation polynomial. Finally, we extend to odd characteristic a result of
Charpin and Kyureghyan (2010) providing an upper bound for the differential
uniformity of the function and its switched version via a trace function.
