  For a map $\varphi\!:U(\mathfrak{g})\rightarrow A$ of associative algebras,
$U(\mathfrak{g})$ the universal enveloping algebra of a (complex)
finite-dimensional reductive Lie algebra, the representation theory of $A$ is
intimately tied to the representation theory of the $A$-subquotient known as
the reduction algebra for $(A,\mathfrak{g}, \varphi)$. Herlemont and Ogievetsky
studied differential reduction algebras for the general linear Lie algebra
$\mathfrak{gl}(n)$ as the algebra of $h$-deformed differential operators formed
from realizations of $\mathfrak{gl}(n)$ in the $N$-fold tensor product of the
$n $th Weyl algebra. In this paper, we further the study of differential
reduction algebras by presenting the symplectic differential reduction algebra
$D\big(\mathfrak{sp}(4)\big)$, by generators and relations, and showing its
connections to Bavula's generalized Weyl algebras (GWAs). In doing so, we
determine a new class of GWAs we call $\textit{skew-affine}$ GWAs, of which
$D\big(\mathfrak{gl}(2)\big)$ and $D\big(\mathfrak{sp}(4)\big)$ are examples.
We conjecture that the differential reduction algebra of the orthosymplectic
Lie superalgebra $\mathfrak{osp}(1|2n)$ is a twisted generalized Weyl algebra
(TGWA) and that the relations for $D\big(\mathfrak{sp}(2n)\big)$ yield
solutions to the dynamical Yang-Baxter equation (DYBE).
