why can consoles only handle lower resolution, settings or manage only 30fps on some games? hardware. what you need to consider is the cheap price of the consoles and the demanding nature of some games running on lower end hardware. while consoles are convenient and can run many games they are not as powerful as computers nor capable of the graphical computation... however to get a decent computer you will need to spend a bit more.
another thing to keep in mind is to make sure you disable 120hz/240hz mode on your tv if you have it as well as motionflow or whatever backlight flickering technology your tv may have to try and smooth out video play. not running the tv at 60hz mode introduces input lag due to the tv post processing (read: faking frames in) which could be affecting how smooth the game feels to you.
as for building a gaming pc under $500, that is going to be tight once you subtract $100 for the os you're now left with $400. factor about $70 for ram. down to $330. at least $50 for a decent psu thats down to $280. $100 for a fair to middling cpu down to $130. motherboard is going to be at least $60 for something low end. down to $60. consider that you need to buy a case, at least $30 which brings you to $30 left to buy a graphics card, mouse and keyboard as well as a hard drive. consider that any graphics card worth buying over a ps4 is going to run about $150, hard drives are at least $50 and you see how your budget is a little light for your typical gaming pc.
if you could bring your budget up to at least $600-650 and find items that are deeply discounted on sale then its quite possible to make something like an fx6300, 8gb ram, r9-280 or r9-270x based system which would perform superior to the ps4.
what you could do for $500 is either an apu based system (processor/graphics integrated on single chip) which is pretty non-upgradeable, running integrated graphics on a low end intel i3, or a g3258 based pentium system with a low end graphics card and cheaping out along the way on components. i will say however that running such low end systems is not going to be an improvement over the ps4 in many cases. see https/www.youtube.com/watch?v=JxUPJdcChzE
both the g3258 and i3 systems are upgradeable though to use better cpu/graphics when you can afford them while a ps4 and apu based system are not.