Both have certain advantages.
-made specifically for games so typically runs fluidly
-control layouts tend to carry from one to the next
-no install and more focus-able help(if there is a problem, your likely not the first)
-if something is wrong inside, parts(as far as I know) are not usually sold individually and some systems are more complicated. Example: In my computer class, we got a client who wanted us to fix their Xbox360, we had to order a kit just to open it.
-do to controller use, number of control and control versatility is downed a bit
-if a certain part is broken, you may have to replace the system entirely
-if PC is built with the proper parts, it can run just s well a the console if not better
-mods and updates are more readily available to add versatility/function to the games
-keyboard=more buttons=more control capability Example: In Grand Fantasia, you can use the number keys to activate skills
-if something is wrong with a part of the computer's hardware/software, you can buy replacement parts as needed/you can download software items to update your software
-if not built right, won't run as well as a console
-installation and set up process can take time
-viruses can cause problems for computer as a whole
-more complicated, certain programs can clash or certain problems may not have easy peesy fixes
I have a Wii, several computer games, and have used my friends Xbox360. I have seen both PCs and consoles in action and would have to say that neither is better. Consoles can run better than PC's because they are designed specifically for the games, but a properly built PC can run better and has more efficient internet capabilities. It mostly depends on your own preferences as both methods also tend to have the same costs. I have CoD:MW3 and much prefer using the Wii remote to aim rather than the 360 controller. I have a VERY difficult time imagining a real time strategy game on a console, but know it's SOMEHOW possible, albeit probably a little difficult.
As with anything, there are certain trade-offs. All you need to do is check what your getting, think about how YOU want to play, buy accordingly, and most importantly, have fun with your games.