I'm not sure that I really know the difference, but I will give you my best guess. To me, a shop is a smaller store, usually privately owned (not a chain store) and only selling a certain type of products. In the US, we often use the phrase "Mom and Pop Shop" which means a small store (usually like a convenient store) which is owned and operated by an older couple who lives nearby. I think a store is anything else. It can be a famous brand or chain.

Shop for definition is a room where tangible products are sold and intangibles like service too. Another definition of this is a place for making or repairing goods, machinery, vehicles, and etc. Ex. for the first definition: "There is a shop nearby that sells emergency stuffs." Ex. for the Second definition: " I am taking my car today to the shop to fix its front bumpers." Store has many definition, Store as in storage and/or to stock for future purposes. And if you want to use the word store just like the Word SHOP, then it can be. But only for sentences with the same meaning for definition no.1 in SHOP. I hope this can help you.

If you are talking about somewhere to buy things, in England we very rarely use the word "store". People would know what you meant, but it would not sound as natural as saying "shop". The main use of the word "store" would be like Rovil said, as a verb, "to store something away". I think maybe "store" is more common in American English? Hope that helps :)