There is not really any difference between Nubia and Kush. They are different names for the same thing. To the extent that there actually is a difference, it is that the name "Kush" was used in earlier times and the name "Nubia" was used later on. Both names refer to an area that was mostly in what is now Egypt and Sudan.
As the links below show us, the words Nubia and Kush refer to the same place. This link tells us that the kingdom was referred to as Kush when its recorded history began in 2613 BC. By the 900s BC, when it conquered Egypt, it was known as Nubia.
So, there is no real difference between these two names; they both refer to the same place.