More specifically, how does a wave-particle duality differ from a quasiparticle/collective excitation?
What makes a photon a gauge boson and a phonon a Nambu–Goldstone boson?
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Not all phonons are Nambu-Goldstone bosons and not all Nambu-Goldstone bosons are phonons. Nambu-Goldstone bosons are (usually) gapless excitations that arise from spontaneous symmetry breaking. For instance, in a spinless Bose-Einstein condensate, the NG boson is indeed a phonon, with a linear dispersion at low energy. However, in a ferromagnet the NG boson is called a magnon. This magnon is gapless but has a quadratic dispersion relation, like a massive particle, and should generally not be called a phonon.
In a periodic crystal, for instance, phonon modes arise because of (discrete) translational symmetry but not spontaneous symmetry breaking -- they are not NG bosons. As someone pointed out, the periodicity is not necessary. In fact, in virtually all condensed matter systems have phonons because of translational symmetry (air has plenty!).
As you can probably tell, I'm much more inclined towards condensed matter than high energy theory, so I'm not sure if I can say anything useful about photons!
Phonons are Goldstone bosons of a spontaneously broken spacetime symmetry (see e.g. http://arxiv.org/abs/hep-ph/9609466). Typically, one breaks Galileo's (or Poincare') boosts and traslations but the resulting number of Goldstones is less or equal than the number of broken generators (e.g. a spacetime dependent traslation is not independent than a boost). Phonons are spin-0 and become strongly coupled, as every Goldstone boson, to the scale (e.g lattice scale) where the underlying microscopic degrees of freedom can be excited.
Massless Spin-1 bosons are instead described by mean of gauge invariance which is a bookkeeping 'symmetry' which erase all interactions that would either give the photon a mass or lower the cutoff of the theory making it strongly coupled at quite large distances. One can try to connect spin-1 massless gauge bosons with Goldstone bosons by breaking spacetime symmetries with a spin-1 order parameter. These are very speculative ideas that have a tiny chance to work only in Poincare' broken spacetimes.