Here is an off the cuff list of advantages to PC gaming over console gaming.
1) Free online. This obviously depends on your console of choice, but not paying monthly fees for gaming on top of my monthly fees for internet service is welcome.
2) More controller variety. While not every single game ever made on PC has significant flexibility in controller input, that still leaves way more flexibility than consoles afford. KB/M offer extreme precision in shooters, but if you get lazy and want to lay back, use a 360 controller for 95% of the games put on Steam in the last 3-4 years (and presumably going forward).
3) Better visuals and image quality. Higher resolution, higher frame rate, less tearing.
4) Better sound. Uncompressed audio.
5) Better multimedia functionality. If you enjoy your Blu Ray, Netflix, Hulu Plus, or ESPN360, imagine taking all those things and putting them on one platform simultaneously (instead of requiring both PS3/360) and also add, on top of this, access to an enormous library of music, a variety of internet browsers, other TV sites like ABC.com, and full Flash/plugin support. In addition, your PC can also be your TIVO: just last night I recorded a movie off my TV directly on to my hard drive. And I didn't need to be there to do it.
6) Larger storage. It's very nice to have literally hundreds of games at your fingertips already installed on your HDD: did the recent release of ME2 on PS3 spark your interest in playing the game again? No problem, it's already right at my fingertips ready to play fully installed for maximum efficiency. This is also true for tv, movies and music: you can have an entire library of films at your fingertips stored directly on your PC.
7) Better online infrastructure. Not only are there multiple online stores -- instead of just one offered on PS3/360 -- but some (well, one) of those stores have superior overall functionality to either PSN or XBL. Again, for free. Don't like Impulse's design, functionality, or library? No problem, try Good old Games. Don't like them? Try the behemoth that is Steam.
8) Mods. These can extend the length of a game's life for you or make a fun game even more fun.
That's it, off the top of my head. You'll notice a common theme amongst these points: power and flexibility. More choices and better performance. What's more, in most cases, you can choose to treat your PC just like a console: play your games with a standard 360 controller and ignore the multimedia features, if you so desire. In other words, it does everything a 360/PS3 does, and more.
This does not mean that everyone should immediately jump up and denounce consoles for all time. Not only does this require a new investment -- or perhaps, a reinvestment, after having already invested in consoles -- but it also requries time and effort. Connecting PCs to home theatre systems is certainly possible now, but not quite as easy and obvious as a console is. Consoles tend to be better streamlined and easier to use. Those qualities are chided on GAF, but are perfectly valid things to like. Not everyone has time to mod, not everyone wants to run HDMI cable through the wall (which is what I did, so that I could have my Desk setup AND connect my PC to the TV simultaneously), not everyone is comfortable tinkering even with seemingly easy settings like resolution/MSAA options.
And that's fine. I'm not telling you your values are dumb if you prefer simplicity and ease of use. People asked what the strengths of the PC are: here you go.