I see what you're saying between the 360 and the One vs the PS3 and 4... The differences in quality as it relates to the upgrade in hardware is more so significant in the Playstation platform vs the Xbox. But that's always been the case, and the bane of Xboxes existence... cheaper internals to keep the price of the box down to compete in the gamers market. Hence the reason why most avid and true gamers may prefer the PS over the xbox (especially as the screen size increases).
However I don't believe its merely a matter of being able to better handle 1080p content more so than the superior quality of the hardware used in the Playstation systems all around. Which determines the quality of the graphics being outputted; 3D rendering, AA, and all that. So in essence 1080p is fine. I'm pretty sure if you play pre-rendered graphics or say a music video or movie side by side, they'd look the same. But people with more information feel free to chime in on the matter.
I sold my console ages ago, reason why I stick to PC gaming, none of these systems could push out the quality of a pretty decent gaming rig anyways.
Owned:
>> Samsung Galaxy Note 3; LG Optimus G; Nexus 4; Samsung Galaxy Ace Duos
<< Microsoft Surface RT; Dell Venue 8; Asus Transformer