It really depends. Yeah, you can build a budget gaming PC, and that is great for older and simple games. However, PS4 titles are also heavily optimized for PS4 hardware and you will be able to play titles on the PS4 which can't be played on your budget gaming PC (well, they'll look like shit at any rate). There are also console exclusives which suck but not much you can do if you want to play them.
I'd rather play on lowest possible settings on 1080p than max 720p.