On PS4, after inserting a disc, you'll be shown a progress bar that ends when you're able to launch the game, and if there's a day-one update, it downloads in the background, and you can install it when you want to play online or whatever the case may be for that update.
On Xbox One, you insert a disc and are shown a progress bar showing the percentage of the full installation, and when enough is installed to run the game, you'll see a message that says "Ready to Start!" The only difference is that the installation from the disc doesn't seem to start until you've downloaded and installed the update.
Now, there's another big difference here. Both consoles will let you know when the game is ready to launch. But once you get into the game, some parts may not be finished installing. For ACIV and Battlefield 4, that would mean the multiplayer modes (on both platforms). However, it seems that Xbox One lets you launch a game even before some singleplayer content is ready. I tested this with Madden 25 on Xbox One. Once the game was "ready to start" most modes like Connected Franchise or Practice were locked off until the game was completely installed. However, after installing the same game on PS4, those modes were playable after the initial install, which took 38 seconds.
