The extra power will probably show itself in the UI and some extra polish. It'll be like the Xbox and PS2. The PS4 will have the best versions of games, only it'll be one the dominant console this time.

Bragging rights? and a better nights sleep for the obsessed.

By now only PS4-tech-demos impressed me. The actual games showed no advantage over Xbone-games, so far. But I think there will be a noticable difference in head2head-comparisons, perhaps already with AC4 or Fifa.
Going to be interesting.

1. Bragging rights
2. ?

It adds new masturbatory scenarios. You ain't a real fanboy until you masturbate to numbers, specs, screenshots and unproven articles.

Higher detail and/or framerate, but it will probably be noticeable almost only in exclusives (so a direct comparison with the same game on another platform won't be possible) or if devs will be willing to put some additional efforts and resources in tweaking each version of a multiplat.
In most cases the improvement even of the weakest from the previous gen will be far greater and noticeable than the differences between two 8th gen versions.