While the abyssopelagic zone shares some physical similarities with the bathypelagic (complete lack of any sunlight, extreme pressures, temps just above freezing), the abyssopelagic zone has some geologic and ecologic distinctions.
Geologically, the abyssopelagic covers the depth that corresponds to the continental slope, while the bathypelagic covers the depth that corresponds to the continental rise. [source: NPS]
Ecologically, the abyssopelagic is a distinctly difficult habitat for organisms. Pressure is even higher and temps slightly lower than the bathypelagic. But the main difference is that in the bathypelagic, consumers like anglerfish can feed off the remains of dead organisms that float down through the water column. By the time you get to the abyssopelagic, most of this source has been used up. Abyssopelagic organisms such as deep sea crabs must search the deep ocean floor on the continental rise to find remains of larger organisms that made it down that far. While the hadopelagic zone is even deeper, it is comprised of ocean trenches which often have hydrothermal vent activity that supports its own chemosynthetic community. See CK-12 Marine Biomes for more info.