  The Gutzwiller wave function solution of the $t$-$J$-$U$ model is considered
for the bilayer high-T$_C$ superconductor by using the so-called diagrammatic
expansion method. The focus is on the influence of the interlayer effects on
the superconducting state. The chosen pairing symmetry is a mixture of
$d_{x^2-y^2}$ symmetry within the layers and the so-called $s^{\pm}$ symmetry
for the interlayer contribution. The analyzed interlayer terms reflect the
interlayer electron hopping, the interlayer exchange coupling, and the
interlayer pair hopping. The obtained results are compared with selected
experimental data corresponding to the copper-based compound Bi-2212 with two
Cu-O planes in the unit cell. For the sake of comparison, selected results for
the case of the bilayer Hubbard model are also provided. This paper complements
our recent results obtained for the single-plane high temperature cuprates [cf.
J. Spa{\l}ek, M. Zegrodnik, and J. Kaczmarczyk, Phys. Rev. B {\bf 95}, 024506
(2017)]
