This question is somewhat tricky, because to a large extent, they do tend to refer to the same thing. According to the University of Chicago, the term Nubia only emerged later as a means of referring to the region, mainly associated with Rome. Kush, on the other hand, was a much older term referring to the same place. According to its webpage:
For much of antiquity, the region south of the 1st cataract of the Nile was called Kush. The name is known from ancient Egyptian, classical and biblical texts. Whether it reflects an indigenous term is not known.
However, with that in mind, it does seem that there are additional levels of nuance within this conversation (especially if we are discussing it from the perspective of modern archaeology and historical scholarship). Janine Bourriau writes:
The King of Kush is the name given in Egyptian sources to the king whose capital lay at Kerma. Archaeologists use Kerma as an adjective to describe the culture of the Kushites and distinguish it from other contemporary Nubian cultures, such as C group and pan grave.
(The Oxford History of Ancient Egypt, ed. Ian Shaw, Oxford University Press: 2000, 196).
In a later chapter from the same book, contributed by editor Ian Shaw, he writes:
The Egyptians' African neighbors to the south included, over the course of time, a number of different ethnic groups in Nubia (primarily the A Group, the C Group, the Kerma civilization, the pan-grave culture, the kingdom of Kush, the Ballana culture, and the Blemmyes) (308).
From this perspective, it seems as if these two terms can also be used as a way to differentiate between the larger geographical region (that being Nubia), and a specific culture which emerged within that region (that being Kush, or rather the Kingdom of Kush).
Even so, Kush exerted a profound and long lasting dominance within the region. Additionally, consider that it was also Kush which carried out the conquest of Egypt. To quote National Geographic:
The legendary Kingdom of Kush, with its series of capitals in what is now northern Sudan, helped define the political and cultural landscape of northeastern Africa for more than a thousand years. ... The Kingdom of Kush is probably the most famous civilization to emerge from Nubia. Three Kushite kingdoms dominated Nubia for more than 3,000 years, with capitals in Kerma, Nepata, and Meroë.
When we think about Ancient Nubia in historical terms, it is almost invariably Kush which we are thinking about.