  The paper reveals clear links between the differential-difference
Kadomtsev-Petviashvili hierarchy and the (continuous) Kadomtsev-Petviashvili
hierarchy, together with their symmetries, Hamiltonian structures and conserved
quantities. They are connected through a uniform continuum limit. For the
differential-difference Kadomtsev-Petviashvili system, we introduce Lax triads
to generate isospectral and non-isospectral flows. This approach provides an
integrable master symmetry and simple zero curvature representations of flows.
The obtained flows are then proved to generate a Lie algebra w.r.t. Lie product
${\llbracket} \cdot, \cdot {\rrbracket}$, which leads to two sets of symmetries
for the isospectral differential-difference Kadomtsev-Petviashvili hierarchy,
and the symmetries generate a Lie algebra, too. The algebra of flows also
provide recursive relations of the flows via the master symmetry, which are
then used to derive Hamiltonian structures for both isospectral and
non-isospectral differential-difference Kadomtsev-Petviashvili hierarchies. The
Hamiltonians generate a Lie algebra w.r.t. the Poisson bracket
$\{\cdot,\cdot\}$. The Hamiltonians together with symmetries lead to two sets
of conserved quantities for the whole isospectral differential-difference
Kadomtsev-Petviashvili hierarchy and they also generate a Lie algebra. All the
obtained algebras have same basic structures. Then, we provide a continuum
limit which is different from Miwa's transformation. By means of defining
\textit{degrees} of some elements with respect to the continuum limit, we find
that the differential-difference Kadomtsev-Petviashvili hierarchies together
with their Lax triads, zero curvature representations and all integrable
characteristics go to their continuous counterparts in the continuum limit. We
also explain the basic structure deformation of Lie algebras in the continuum
limit.
