  Flux-limited X-ray samples indicate that about half of rich galaxy clusters
have cool cores. Why do only some clusters have cool cores while others do not?
In this paper, cosmological N-body + Eulerian hydrodynamic simulations,
including radiative cooling and heating, are used to address this question as
we examine the formation and evolution of cool core (CC) and non-cool core
(NCC) clusters. These adaptive mesh refinement simulations produce both CC and
NCC clusters in the same volume. They have a peak resolution of 15.6 h^{-1} kpc
within a (256 h^{-1} Mpc)^3 box. Our simulations suggest that there are
important evolutionary differences between CC clusters and their NCC
counterparts. Many of the numerical CC clusters accreted mass more slowly over
time and grew enhanced cool cores via hierarchical mergers; when late major
mergers occurred, the CC's survived the collisions. By contrast, NCC clusters
experienced major mergers early in their evolution that destroyed embryonic
cool cores and produced conditions that prevented CC re-formation. As a result,
our simulations predict observationally testable distinctions in the properties
of CC and NCC beyond the core regions in clusters. In particular, we find
differences between CC versus NCC clusters in the shapes of X-ray surface
brightness profiles, between the temperatures and hardness ratios beyond the
cores, between the distribution of masses, and between their supercluster
environs. It also appears that CC clusters are no closer to hydrostatic
equilibrium than NCC clusters, an issue important for precision cosmology
measurements.
