From what I've seen, perfect pitch literally just means you have a different process for orienting yourself in pitch-space than people who mainly rely on relative pitch processing. A relative pitch process would look something like this: Note 1, go up a 5th, down a major 3rd (outlining a minor triad), then down a minor 2nd, and finally up a tritone – for scale degrees, this would be 1, 5, minor 3, 2, minor 6. If someone told you that the tonic was G, then you could put labels to this.
Meanwhile, perfect pitch processing without relative pitch context would be G, D, B flat, A, E flat – and then from this, they could extrapolate scale degrees and intervals, but that's less of an automatic process than it seems to be for a lot of people who primarily rely on relative pitch processing. Overall, some things are easier, while others are a lot less intuitive.