Take for example, the Abyssal Greater Basilisk. Is this just an advanced Basilisk with the Fiendish template or is there more to being an Abyssal monster?
The fiendish template can make a creature into any kind of fiend, that is, an outsider from any of the Lower Planes, of any alignment.
“Abyssal” specifies that the creature is specifically a Chaotic Evil fiend from the “Infinite Layers of the Abyss” as Monster Manual puts it. “Infernal” would be similar for Hell. There isn’t really a consistent adjective used for Gehenna, Hades, or Carceri.
Aside from that, they produced the stats for the abyssal greater basilisk by advancing its Hit Dice and applying the fiendish template, and “abyssal” is just descriptive of the alignment and environment they assigned it.