In both standard written and colloquial British English it's almost always "shop", but the term "store" is normal in the jargon used by people in the retail business. It's quite common to hear the word "store" used in a public announcement in a shop, for example.

There are many things like that. For example, normal people say "tin", in my experience, but the signs in a shop seem to use "can" more often than "tin", bizarrely. You'd probably find many similar example if you walked round some UK shops looking at all the labels.

There's also a whole work/office jargon used in modern Britain: words and grammar that people use at work but wouldn't normally use at home or in the pub.

In general I don't like jargon. To me it seems both pretentious and small-minded. On the other hand, an obscure technical term, used correctly, is like a beam of sunlight or a breath of fresh air. To me, at least. Though they sometimes borrow from each other, jargon and technical vocabulary are very different things.