Digital Foundry has some initial conclusions from its testing of the game, and both the PlayStation 4 and Xbox One run the game at a full 1080p resolution. The game looks very similar across both platforms, in fact.
"Texture maps are matched for resolution, with a generous level of anisotropic filtering across the ground for good measure," the post states. "Each [console] uses the same grade of screen-space ambient occlusion, to match PC's highest, and effects quality is identical too. With everything being so close in the visual stakes though, how does the frame-rate on these machines hold up?"
The full post is worth reading, but the short version is that both consoles struggle with maintaining 30fps throughout the game. The main difference is the Xbox One has a stuttering issue that isn't present on the PlayStation 4.
"It's a glaring hitch downward, and matching runs to the gates of Diamond City shows Xbox One dropping to a record 0fps (zero) while PS4 turns the same corner at 28fps," the article explains. "Each has their blips, but having tested two separate Xbox One and PS4 consoles, the results are always the same across the world at large; we get sizeable stutters on Microsoft's console that aren't present on PS4."
There are areas where the PlayStation 4 suffers in relation to the Xbox One, but the overall consensus is you should stick to Sony's platform if possible, and if your PC isn't up to the task. "Evidence so far suggests those buying4 on console should get an overall smoother experience from PS4. It's far from perfect, and we have a suspicion re-tests will be due once each machine receives new patches down the line."