Potential density
Potential density is a dynamically important property: for static stability potential density must decrease upward. If it doesn't, a fluid parcel displaced upward finds itself lighter than its neighbors, and continues to move upward; similarly, a fluid parcel displaced downward would be heavier than its neighbors. This is true even if the density of the fluid decreases upward. In stable conditions (potential density decreasing upward) motion along surfaces of constant potential density (isopycnals) is energetically favored over flow across these surfaces (diapycnal flow), so most of the motion within a 3-D geophysical fluid takes place along these 2-D surfaces.
Vertical pressure variation
Whether or not density and gravity can be reasonably approximated as constant depends on the level of accuracy needed, but also on the length scale of height difference, as gravity and density also decrease with higher elevation. For density in particular, the fluid in question is also relevant; seawater, for example, is considered an incompressible fluid; its density can vary with height, but much less significantly than that of air. Thus water's density can be more reasonably approximated as constant than that of air, and given the same height difference, the pressure differences in water are approximately equal at any height.