In two schools of Buddhism, an important part of the practice is the repetition of special mantras which sound like simple expressions of honor or gratitude to those of us outside of the sects, but which have profound meaning to the practitioners. Is it possible to explain the deeper meaning of Nam-myoho-renge-kyo ("Homage to the Lotus Sutra") as Nichiren Buddhists understand it or Namu Amida Butsu ("Homage to Amida Buddha") as Jodo Shinshu Buddhists understand it or must these thing be understood only through experiencing them?
My answer is from the Shin perspective, I'll let someone else answer for Nichiren. Indeed you're right in saying that there are "two" meanings of Namu-Amida-Butsu - and that for the person of faith, it has a special meaning. However, in Shin, Namu-Amida-Butsu is not a mantra. It is understood within an entirely different framework from that of Esoteric mantras. For a person without the mind of faith (jp.: shinjin), "Namu-Amida-Butsu" is an effort of calling to Amida for salvation. As you probably already know, the Original Vow of Amida is that people who will sincerely and joyfully entrust themselves to Amida and call upon his Name, they will be saved and will be reborn into the Pure Land. So for a person without shinjin (which means having no doubt whatsoever about Amida's Vow), "Namu-Amida-Butsu" might still be a practice like others in Buddhism, an effort of self-power. For such a person, calling Namu-Amida-Butsu is a willingness to rely on the Other-Power of Amida (on his practice of saving sentient beings). But from the perspective of a person of shinjin, a true person of faith, chanting or reciting "Namu-Amida-Butsu" thinking that this is what will get you to the Pure Land is still caught within the logic of calculating mind, calculating merit ("the more you say it, the more chance you will have to be reborn in the Pure Land"). This mind that "does religious deeds" is the very mind of self-attachment! For a person of shinjin - for Shinran - this logic has to be absolutely abandoned. "Namu-Amida-Butsu" is then understood entirely differently: not as something we do, not as our own practice, but as an embodiment of Amida's practice. Another way to put it, once one totally abandons the idea that it's possible to "save yourself" through enough effort, one totally relies on Other-Power and so there is nothing to practice anymore on our part. There's just an absolute passivity of "deep listening to the Dharma". One realizes that Amida already accomplished all practices so there's nothing to do on our part except to listen to this truth and to rely on it fully. This reliance is true faith, the diamondlike mind, and "Namu-Amida-Butsu" is just the verbal expression thereof. It's the "linguistic embodiment" of our already-having-been-saved by Amida. One is then only a witness of this salvation event, not someone who is practicing to achieve one's own salvation. To practice anything after the salvation-event is to not understand that event. Practising anything after having been saved by Amida would be like looking for a job after you've already won $10 million in the lottery, to give a mundane example.
I realize I didn't do a good job at explaining this here ... it's hard to do it in this format, but I hope that at least it became clear that the Pure Land Way of Master Shinran has nothing to do with Esoteric Buddhism or Mantrayana / the Way of Mantras. It's a wholly different paradigm.