This website is a pretty great place to look at to get your first glimpses of an XBONE game vs. a PS4 game:
TL;DR is basically that both XBONE and PS4 versions of the game isn't as good as the PC on the highest setting. It's worth pointing out that it was captured using dev kits and the games at a near-finalized state. Also, XBONE is a couple of weeks outside of the PS4 release so who knows.
Battlefield 4's performance is almost perfectly handled in campaign mode, with the avoidance of full 1080p allowing both PS4 and Xbox One to deliver smooth 60fps playback with v-sync engaged. Exceptions are few and far between, with drops to 50fps seen in three cut-scenes here, and once down to 30fps during an explosive battle.
And also, PS4 multiplayer's Framerate drops a bit also, but there's not XBONE footage to compare it with.