  We compare the bottom of the spectrum of discrete and continuous
Schr\"odinger operators with periodic potentials with barriers at the
boundaries of their fundamental domains. Our results show that these energy
levels coincide in the semiclassical limit and we provide an explicit rate of
convergence. We demonstrate the optimality of our results by using
Bohr-Sommerfeld quantization conditions for potentials exhibiting
non-degenerate wells, and by numerical experiments for more general potentials.
We also investigate the dependence of the spectrum of the discrete
semiclassical Schr\"odinger operator on the semiclassical parameter $h$ and
show that it can be discontinuous.
