In this post I am using phaser ce 2.11.1 of phaser.
On use case example of game.time.elapse would be an example that moves a sprite by a pixel per second rate. This helps to ensure that a sprite will always move as the same rate regardless of how fast or slow a device might be, and the value is also automatically adjusted by phaser when the project goes inactive due to a pause of inactive windows event.
In this method I am making a sprite and setting a pixel per second value to the data object of the sprite. I am also creating an update method for the sprite that will use game.time.elapsed to move the sprite. In addition I am also making use of Phaser Wrap method to make the sprite wrap back around to the beginning of the game world when it goes out of bounds.
This method uses canvas to create a simple sprite sheet for the example.
Now to pull it all together with a Phaser.Game instance
So game.time.elapsed is a great way to move a display object by a pixel per second rate commpared to other methods so that the sprite moves at the same rate no matter what. In addition it is effected by game pause and inactivity events so it is better than just going by the current time alone.