If you spend however much the PS4 will cost 400-600 dollars, you WILL get more performance than the PS4 offers. Especially when you consider that if you feel like it, you can upgrade/change parts later if you want. You can also buy the desktop now, and buy a PS4 a few years later when its cheap and "old".
Usually games are also cheaper for the PC, especially after a little while. Right now, you can get Alan Wake for 1$ for the computers. Steam sales are very abundant with pretty much -75% sales twice a year and more. You will also have a larger selection of games, full backwards compatability etc.
Ofcourse, you can do MORE with that gaming pc than just game. Ofcourse you can browse the internet, watch videos and movies, write essays and do other school/work related things. So a PC is more than a console.
TL;DR Gaming desktop is a better choice.