PRISM (Polarized Radiation Imaging and Spectroscopy Mission) was proposed to ESA in May 2013 as a large-class mission for investigating within the framework of the ESA Cosmic Vision program a set of important scientific questions that require high resolution, high sensitivity, full-sky observations of the sky emission at wavelengths ranging from millimeter-wave to the far-infrared. PRISM's main objective is to explore the distant universe, probing cosmic history from very early times until now as well as the structures, distribution of matter, and velocity flows throughout our Hubble volume. PRISM will survey the full sky in a large number of frequency bands in both intensity and polarization and will measure the absolute spectrum of sky emission more than three orders of magnitude better than COBE FIRAS. The data obtained will allow us to precisely measure the absolute sky brightness and polarization of all the components of the sky emission in the observed frequency range, separating the primordial and extragalactic components cleanly from the galactic and zodiacal light emissions. The aim of this Extended White Paper is to provide a more detailed overview of the highlights of the new science that will be made possible by PRISM, which include: (1) the ultimate galaxy cluster survey using the Sunyaev-Zeldovich (SZ) effect, detecting approximately 106 clusters extending to large redshift, including a characterization of the gas temperature of the brightest ones (through the relativistic corrections to the classic SZ template) as well as a peculiar velocity survey using the kinetic SZ effect that comprises our entire Hubble volume; (2) a detailed characterization of the properties and evolution of dusty galaxies, where the most of the star formation in the universe took place, the faintest population of which constitute the diffuse CIB (Cosmic Infrared Background); (3) a characterization of the B modes from primordial gravity waves generated during inflation and from gravitational lensing, as well as the ultimate search for primordial non-Gaussianity using CMB polarization, which is less contaminated by foregrounds on small scales than the temperature anisotropies; (4) a search for distortions from a perfect blackbody spectrum, which include some nearly certain signals and others that are more speculative but more informative; and (5) a study of the role of the magnetic field in star formation and its interaction with other components of the interstellar medium of our Galaxy. These are but a few of the highlights presented here along with a description of the proposed instrument.