We investigate bottomonia splittings by solving a Schr\"odinger-Pauli-type equation with parametrizations of QCD potentials around those that have been determined previously in lattice simulations. This is done both in the continuum and on finite lattices with resolutions ranging from $a=0.2\mathrm{fm}$ down to $a=0.025\mathrm{fm}$ and an extent of up to 12 fm or ${144}^{3}$ lattice points. We find a strong dependence of some splittings, in particular the $2S\ensuremath{-}1S$ and $1P\ensuremath{-}1S$ splittings, on both the quark mass and the short range form of the static potential in the neighborhood of the b quark mass, while splittings such as $3S\ensuremath{-}2S$ and $2P\ensuremath{-}2S$ show reduced dependence on the short distance potential. We conclude that the quenched quarkonium spectrum cannot be matched to experiment with a simple redefinition of the lattice spacing. We investigate the size of relativistic corrections as a function of the quark mass. Finite size effects are shown to die out rather rapidly as the volume is increased, and we demonstrate the restoration of rotational symmetry as the continuum limit is taken.