Imagine, I asked you to sing an interval, for ex. G – C#. How would you do it? I would have to give you the first tone (G), then you’d construct a tritone in your head and then you’ll be able to sing the C#, right?
With absolute pitch you know instantly and can sing the G, then the C#. So, you omit the mental construction and comparison.
At the moment, your Relative Pitch is overloaded because you are using your logical, for most people left, side of the brain to do several operations, one at a time: keep track on the 1 comparison tone, doing actual comparison (here, the right side of the brain comes into the play since it has to recognize a pattern, but then the left side takes over again as the recognized pattern has to be compared to the “installed” interval prototype), calculating the second tone.
For a well trained musician this process is fast, yet, it is observable in speed (not instant) AND demanding in terms of concentrating on the 1st rate “arithmetical” calculation. Often, one has to do the second rate calculation at the same time (functional degree analysis, that is, calculating harmonies).
As you begin using your absolute pitch, you gradually release yourself from the 1st degree calculation since you do not do any comparison. You RECOGNIZE, audiolize and recall (thus can reproduce) a sound as a C, an F or whatever pitch it is. Just as you recognize a face of a friend, instantly.
The amount of energy which you spend now on the 1st degree of relative pitch analysis, when released, can be used for more advanced things, like for example comprehensive analysis of the composition on the whole or analysis of the compositional language, or analysis of the orchestration. Just now, you hardly have time and energy for all these things.
For example, I used to think that Dave Brubeck’s music was complicated when I only had to go by the relative pitch. Now, when my AP took over most of that unnecessary work, I see that this music is very transparent and simple in terms of harmonies. (This does not make me appreciate the beauty of the music less than before!)
The way M.A.P. method is build, you will train yourself not to switch from relative pitch to absolute pitch, but to make the two go together.