The biggest drawbacks for me are threefold:

Cost - This has already been explained by others. Yes, you do get a PC that can be used for other things out of the deal but personally I don't like sitting at a PC and browsing online. I usually do that from a laptop or tablet, which aren't really great for hardcore gaming. So for me a PC would strictly be a gaming device and therefore the initial cost is more than a console.

Convenience - A lot of this stuff is minor. A console is more plug and play, but these days not really by much. There's also the aspect of the console being a 'complete' product that doesn't require you to do anything else to it, while a PC may need some maintenance, upgrading and knowledge of what's in it. I like putting the disc in my system and then playing it on my couch a minute later, which you can do on PC with some effort, but I hate effort.

Exclusives - This goes with selecting any console, you'll only have the games that are released for it. There's no Mario, Zelda, Halo, Uncharted, FF, etc. for PC. There are many PC exclusives for sure. It just depends on your gaming tastes. I've always loved the Sony and Nintendo exclusives which is why I stick with those systems.