PS4 Pros: Cheap, simple, streamlined hardware
PS4 cons: lower max resolution (1080p), difficult and limited modding (needs to be hacked I believe), many games locked at 30 frames per second, some do go to 60 fps.
Gaming PC Pros: Much more powerful, capable of much higher resolutions (4K for gaming with best hardware), relatively easy modding with huge modding community, with the right hardware can run games at very high framerates (60-120 fps+), also able to do stuff like Microsoft office and browse the web way easier, bigger game library
Gaming PC Cons: A decent gaming PC is going to cost more ~$700-$1,000 including a monitor, games settings need to be tuned for specific hardware, bigger power draw.
Keep in mind you can save some money by using your TV as a monitor and connecting it to your GPU via HDMI. If you are really strapped for cash I would recommend this:
http://pcpartpicker.com/user/bohemond1980/saved/pJcmP6 for AMD processor or this for the absolute cheapest and still decent:http://pcpartpicker.com/user/bohemond1980/saved/KbbdnQ. I wouldn't go below an R7 265/HD 7850 for a GPU.
I use my Xbox One for casual gaming at bedtime or with my nephews. For any serious gaming I use my desktop, consoles just don't compare to a good gaming pc. Control peripherals for PC are also much better in my opinion (mouse, keyboard, joystick), though you can still use controllers if you want. Steam's game library demolishes the releases on both consoles, definitely something to consider.