PC Advantages:
- Higher resolutions and framerates and extra eyecandy if you have the graphics hardware that can handle it.
- More control options, eg. mouse, keyboard, controller, racing wheel, joystick, whatever you want.
- Backwards compatibility, unlike the new consoles which either don't have it (PS4) or where it will be patched in later for a limited selection of games (Xbox One) PC can play games made more than two years ago without you needing to spend more money on a 'remastered' version of the old game. This backwards compatibility can go all the way back to the 1980s with the use of DOSBox.
- No subscription fee to play online (unless you play World of Warcraft or one of the few other sub based MMOs still out there)
- Game genres that are either not present at all on consoles or only very poorly represented eg. Real Time Strategy, 4X Strategy, MMOs, MOBAs.
Console Advantages:
- Exclusive games
- Genres that aren't well represented on PC, namely JRPGs, Fighting Games, Spectacle Fighters.
- Lower cost of entry (though perhaps not lower overall cost when you factor in subscription fees and likely paying more for your games)
- No need to upgrade or replace your console while the current console generation lasts, barring a repeat of the Red Ring of Death or some other design flaw that kills your console shortly after the warranty period has expired.