Personally, I've always favoured PCs, despite not always being able to run the latest games, and although I've never owned a console that doesn't mean that most aren't console nuts.
I agree with many points raised so far, so I'll just try to add what hasn't been mentioned.
The advantages of PCs over consoles have fluctuated over time, whereas the advantage of consoles over PCs has remained fairly constant.
Consoles have that pick up and play nature, don't require a seperate monitor (assuming people already have a TV), there's not the compatibility issues, and other points people have raised. Although gone are the days where you had to knobble your autoexec.bat and config.sys for individual games and reboot, or run seperate text based config utilities in a fight to get sound working. PCs have evolved a lot too.
PCs, back at the start, I feel offered no real advantage at all. As games progressed, variations limitations of the consoles have been reached. Complex games requiring lots of controls don't work well without a keyboard. From the mid 90s, the resolution of PC games steadily started increasing as monitors and graphics cards improved. It's taken a whole decade but finally, thanks to HDTV, consoles have finally caught up in that regard.
The location was mentioned as being more social for a consoles, but I see PCs becoming more usual to live in the front room as time goes on, so consoles could lose that "advantage".
Patching is a good issue already mentioned, it's good that consoles have caught up in this regards, similarly with online play.
The fixed hardware, while having clear advantages, also has downsides. This only applies to games that are being made for PC as well, and more-so where PC is the lead title (certainly rare now), but open world style games (and tbh modern games in general) are very memory thirsty, and the fixed hardware of the console could be just slightly too little to fit in everything you want in your game. PC hardware being so modular, it's easy just to make the game you want and slap a higher memory requirement on the box.
User generated content has become massive in the past years, something else that consoles are over a decade behind on. I don't see any easy solution to this anytime soon, especially given the previous paragraph.
There are other, less usual benefits to PC gaming. Try getting a triple monitor setup going on a console for instance. Or motion simulators. Or dare I say it: Insim.
Backwards compatibility is another console-weak area, although Sony have made some improvements here. Also the ability to join in and beta test games in development, LFS being the obvious example, but open source ventures like OpenTTD (and no doubt many others) allow you to particpate in a way console gaming never will.