  We study the Mott metal-insulator transition in the two-band Hubbard model
with different hopping amplitudes $t_1$ and $t_2$ for the two orbitals on the
two-dimensional square lattice by using {\it non-magnetic} variational wave
functions, similarly to what has been considered in the limit of infinite
dimensions by dynamical mean-field theory. We work out the phase diagram at
half filling (i.e., two electrons per site) as a function of $R=t_2/t_1$ and
the on-site Coulomb repulsion $U$, for two values of the Hund's coupling $J=0$
and $J/U=0.1$. Our results are in good agreement with previous dynamical
mean-field theory calculations, demonstrating that the non-magnetic phase
diagram is only slightly modified from infinite to two spatial dimensions.
Three phases are present: a metallic one, for small values of $U$, where both
orbitals are itinerant; a Mott insulator, for large values of $U$, where both
orbitals are localized because of the Coulomb repulsion; and the so-called
orbital-selective Mott insulator (OSMI), for small values of $R$ and
intermediate $U$'s, where one orbital is localized while the other one is still
itinerant. The effect of the Hund's coupling is two-fold: on one side, it
favors the full Mott phase over the OSMI; on the other side, it stabilizes the
OSMI at larger values of $R$.
