Here we go:
Is it simply a question of GB/US? No, not really. Sorry, but the answer is not that simple.
As usual, the distinction is almost never as straightforward as a 1:1 split between American and British English. Just as it is completely inaccurate to say "Film is British and movie is American" ( in fact both 'movie' and 'film' are used on both sides of the Atlantic) it is (a) simplistic and (b) misleading to say that 'shop' is British and 'store' is American.
Basically, they are both English words. That's it - English. Remember that there is only ONE English language. The difference is just that one used MORE OFTEN by US speakers and the other is used more often by British English speakers.
Both words are used by virtually all English speakers. Speakers of US English use the term 'store' most of the time, but - if I'm not mistaken - they may use the word 'shop' for a very small, local outlet, especially if it specialises in one product or service. Meanwhile British English speakers use 'shop' far more than US speakers do, but it they also use the word 'store' for a larger outlet, and in combinations such as 'department store', 'furniture store' and 'convenience store'. The reality is that it's a question of preference and frequency of usage - not a simple split according to nationality.