ExtremeTech tested NBA 2K14, and their results were similar to the OP ("instant" vs 6 minutes from the first time the disc is inserted. Once the game was installed, the differences were only a few seconds (with the XBox One a little faster than the PS4).
PS4 vs. Xbox One speed test results
Cold boot to main dashboard –
PS4: 20 seconds — Xbox One: 1 minute 5 seconds
Standby/hibernate to dashboard –
PS4: 3 seconds — Xbox One: 12 seconds
Installing NBA 2K14 from disc –
PS4: 36 minutes – Xbox One: 46 minutes (see note below)
Time for NBA 2K14 to be playable after inserting disc –
PS4: Instant — Xbox One: 6 minutes
Time to load NBA 2K14 main menu from the dashboard –
PS4: 38.7 seconds — Xbox One: 29.6 seconds
Time to load a playable Quick Match in NBA 2K14 from main menu –
PS4: 37.6 seconds — Xbox One: 31.4 seconds
Time to switch between playable game, to the dashboard, and back to the game –
Negligible difference between the PS4 and Xbox One. Both are almost instantaneous.
http://www.extremetech.com/gaming/171469-ps4-vs-xbox-one-side-by-side-speed-test-to-decide-which-console-is-faster-video









