We present a detailed X-ray study of the intracluster medium (ICM) of the nearby cool-core galaxy cluster Abell 478 (z - 0.088), based on Chandra and XMM-Newton observations. Using a wavelet-smoothing hardness analysis, we derive detailed temperature maps of A478, revealing a surprising amount of temperature structure for an apparently well-relaxed cluster. We find the broadband Chandra spectral fits yield temperatures that are significantly hotter than those obtained with XMM-Newton, 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 multitemperature spectra and fitting them with a single-temperature model, we find no significant difference between Chandra and XMM-Newton, indicating that nonisothermality cannot fully explain the discrepancy. We have discovered four hot spots located between 30 and 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 c 1) x 10 super(59) ergs. 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 of less than 10% of the thermal excess in the hot spots. The presence of these hot spots could indicate strong-shock heating of the intracluster medium from the central radio source, one of the first such detections in a cool-core cluster. Using the high resolution of Chandra, we probe the mass distribution in the core and find it to be characterized by a logarithmic slope of -0.35 c 0.22, which is significantly flatter than an NFW (Navarro, Frenk, and White) cusp of -1 and consistent with recent strong-lensing results for a number of clusters.