They're the same phenomenon, just different amounts and precision of memory.

Source: my wife and multiple of her siblings have absolute pitch and we've spent hours and hours talking about their experience with it.

Oh, is absolute pitch the ability to recall every note?

I once heard a C on a piano and played it a few times and then remembered it. I thought it was absolute pitch.

Perfect pitch is being able to name any individual note and to hear when they're not quite in tune. Although it's also a curse, because perceived pitch changes with age, so you have to keep learning to compensate. And domestic acoustic pianos are often tuned a semitone flat, which really messes with your head.

Advanced pitch recognition is being able to pick out the notes in a complex chord in open form.

Professional pitch recognition - as demonstrated by some conductors - is being able to list the pitches in any random set of notes played at the same time on a piano. Including notes in the lowest octave.

It's a lot easier to get to the last two if you have the first.