What is meant by Namu Amida Butsu and also Nam-myoho-renge-kyo?

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?