  We clarify the interplay and competition between disorder and flat band in
the Creutz ladder with inter-particle interactions focusing on the system's
dynamics. Without disorder, the Creutz ladder exhibits flat-band many-body
localization (FMBL). In this work, we find that disorder generates drastic
effects on the system, i.e., addition of it induces a thermal phase first and
further increase of it leads the system to the conventional many-body-localized
(MBL) phase. The competition gives novel localization properties and
unconventional quench dynamics to the system. We first draw the global sketch
of the localization phase diagram by focusing on the two-particle system. The
thermal phase intrudes between the FMBL and MBL phases, the regime of which
depends on the strength of disorder and interactions. Based on the two-particle
phase diagram and the properties of the quench dynamics, we further investigate
finite-filling cases in detail. At finite-filling fractions, we again find that
the interplay/competition between the interactions and disorder in the original
flat-band Creutz ladder induces thermal phase, which separates the FMBL and MBL
phases. We also verify that the time evolution of the system coincides with the
static phase diagrams. For suitable fillings, the conservation of the
initial-state information and low-growth entanglement dynamics are also
observed. These properties depend on the strength of disorder and interactions.
