How do I store the time between the LED coming on
The LED doesn't "come on". You turn it on. Surely, you know where in the code you do that. Surely you can call millis() to record WHEN you do that.
and button being pressed.
When the switch is pressed, and when the Arduino sees that the switch is pressed, may NOT be the same time. The state change detection example shows how to determine that the switch HAS BECOME pressed (or released). That would be a reasonable place to add some code to record WHEN that happens.
Now that you have two bits of code, to record two times, it doesn't take a degree in advanced calculus to determine the interval between the times.