Whew, that is a hard one to describe...I´ll give explaining it a go, but I hope someone else chimes in on that. After all I might soon go on to study English (for the record I don´t watch my spelling around here xD), but it is still my second language and I recon even for native speakers these terms are hard to elaborate on.
To try and get my point across I´ll have to go a bit overboard here, so take whatever picture you get and take it down a notch, I am just trying to get the idea of dryness across ;)
The AM has a very direct sound, unforgiving, not processed. There is not that much of a juicyness you´ll find in most other PUs to varying degrees. It is there, bäm, front row center, spotlight, no makeup. No distance, nothing coping with mistakes or imperfections, just brutal honesty. Surgical precision for better and for worse.
It has a tad more of the character you get when you DI record an electric guitar.
Pair that with a good and cooking tube amp and you get a sound that is brutally honest, direct to your face, but not dead or sterile.
To use a metaphor the Nailbombs are this big russian MMA fighter, who has been trained in wrestling, boxing and the likes, but is powered by a lot of rage, ripping and destroying his enemys in a fury, running em threw and pummeling them while they are on the ground. The AM on the other hand is more of a asian martial artist, trained in the likes of Kung Fu and Jiu Jitsu, he is cold, calculating and effective, stricking his enemys down with a single aimed blow.
Hope this helps to illustrate dryness and get the difference between the NBs and the AM across.
Also keep in mind though that the AM into a proper cooking tube amp still is organic, agressive and such, it´s no cyborg or something. Still human, but one in the way I described above.
Oh and the clips posted by guitarIv also give a good idea, though both PUs of course with the rest of the gear can also steered in different directions.