A cognitive process involves thinking. Because we APPs give a note name to the sounds we hear, we have to link that association.
When I recall a melody, I feel like I'm always translating them into their distinct attributes (so this would be like always translating them into note names) so when I recall it, I think of those attributes and remember the order they were in. At least I think that's slower, but it's hard to describe!
I can't name every individual tone because I haven't gone that far, but 'G' seems optimistic, 'A' is overconfident, 'Eb' is a nasal and annoying sound. 'C' sounds neutral. 'F' can be a bit lazy. 'F#' I associate with the word yearning. *shrug*
I'm starting to confuse myself, so please take what I said with a grain of salt!
Relative pitch seems like understanding what was just said and replying normally.
For a musician I don't actually listen to a lot of music so I haven't experienced enough in order to assign something to every note and key.
Sometimes I need to read up on the background of the piece or the intention of the composer to fully understand the piece and its keys and then I'll make those associations.
Even when I'm listening to music sometimes I can sit there and wonder 'What's this supposed to mean?' throughout the whole piece.
I can transpose on the spot (well, mostly!) because of my RP - I am able to hear how the chords and notes relate to each other, and reconstruct those relationships in any key I want to.
Most people without AP, at least that I interact with on a daily basis, talk about music in terms of relative pitch - like, "wow, that was a great bVII7 chord that guy just threw in there".
Basically, what I tend to use AP for now is transcribing without checking what key I'm in, and identifying keys of pieces for people who don't know where they want to sing/play them, or keys of tunes on the radio or something.
My RP isn't as developed as I'd like it to be, but it's getting there. I'm lucky that I can back it up with AP, but RP is certainly the much more desirable skill.