  We present a detailed X-ray study of the intracluster medium (ICM) of the
nearby, cool-core galaxy cluster Abell 478, with Chandra and XMM observations.
Using a wavelet smoothing hardness analysis, we derive detailed temperature
maps of A478, revealing a surprising amount of temperature structure. The broad
band Chandra spectral fits yield temperatures which are significantly hotter
than those from XMM, but the Fe ionization temperature shows good agreement. We
show that the temperature discrepancy is slightly reduced when comparing
spectra from regions selected to enclose nearly isothermal gas. However, by
simulating multi-temperature spectra and fitting them with a single temperature
model, we find no significant difference between Chandra and XMM, indicating
that non-isothermality cannot fully explain the discrepancy. We have discovered
4 hot spots located between 30--50 kpc from the cluster center, where the gas
temperature is roughly a factor of 2 higher than in the surrounding material.
We estimate the combined excess thermal energy present in these hot spots to be
(3+/-1)x10^59 erg. The location of and amount of excess energy present in the
hot spots are suggestive of a common origin within the cluster core, which
hosts an active galactic nucleus. This cluster also possesses a pair of X-ray
cavities coincident with weak radio lobes, as reported in a previous analysis,
with an associated energy <10% of the thermal excess in the hot spots. The
presence of these hot spots could indicate strong-shock heating of the ICM from
the central radio source -- one of the first such detections in a cool core
cluster. We also probe the mass distribution in the core and find it to be
characterized by a logarithmic slope of -0.35+/-0.22, which is significantly
flatter than an NFW cusp of -1. (abridged)
