The rulebook contains the full rules for the game, including character creation, gameplay, and copious notes for running the game--extra rules about pacing and determining challenge difficulty and a suite of tables to enable quick scenario/location/clue/henchman/NPC generation. You and your game group only need one copy of the rulebook to play the game.
The player's guide, on the other hand, contains only the rules for character creation and for playing the game, along with several copies of the character sheet and a space for notes. The player's guide is non-essential, but useful--your players can look up the rules they need themselves. Plus they're fun "zine" type things so everyone will feel extra cool. The player's guide is not essential for playing the game.