What defines cool jazz as a jazz style/sub-genre?
Cool jazz is a style of jazz that arose as a sort of response to the growing popularity in bebop. Bebop featured fast-paced, syncopated, energetic music and frantic improvised solos. Cool jazz, in contrast, had a mellow and emotionally subtle "coolness" about it. There is still much melodic and harmonic complexity to it but you still won't see the same use of diminished and augmented chords that you see in bebop. Cool jazz often incorporates elements of classical music and is usually formally composed as opposed to being largely improvised. Another defining characteristic might be a musician's tendency to play behind the beat instead of driving it. Best examples: Dave Brubeck's Time Out, Bill Evans Trio's Explorations, and Miles Davis's Birth of the Cool.
*Birth of the Cool* might be an even better Miles record to cite here, from a historical perspective. It's easier to place the later, super-laid-back sound of *Kind of Blue* in context when you hear the progression from bop and big-band into the early cool sound, where you have a lot of formal arrangements; and then into the later cool sound, where arrangements get less formal and you see more harmonic experimentation.