  The two major discrete time formulations for quantum walks, coined and
scattering, are unitarily equivalent for arbitrary position dependent
transition amplitudes and any topology (PRA {\bf 80}, 052301 (2009)). Although
the proof explicit describes the mapping obtention, its high technicality may
hinder relevant physical aspects involved in the equivalence. Discussing
concrete examples -- the most general constructions for the line, square and
honeycomb lattices -- here we unveil the similarities and differences of these
two versions of quantum walks. We moreover show how to derive the dynamics of
one from the other by means of proper projections. We perform calculations for
different probability amplitudes like, Hadamard, Grover, Discrete Fourier
Transform and the uncommon in the area (but interesting) Discrete Hartley
Transform, comparing the evolutions. Our study illustrates the models
interplay, an important issue for implementations and applications of such
systems.
