While both consoles will run the game at 60 fps, the PS4 will run it at a native 1080p while Xbox One will render it at 720p. Both last generation versions (PS3 & Xbox 360) will run it at 720p.
Though the gamers are taking this seriously, Hideo Kojima isn’t very concerned about this controversy and even talks about the differences of the game’s different versions with an open and relaxed attitude, which is pretty rare thing to see in a developer of his caliber.
At the bottom of this post you can see, in an animated gif, the differences between the PS4 and Xbox One versions of the title, as well as the PS3 and Xbox 360.
PS4 vs Xbox One
PS3 vs Xbox 360