There's no real difference. In the United States we tend to say "store" more often than "shop" but we say both. In the U.K., I _think_ they use "shop" more often, but I'm not sure. In the U.S. "shop" is a slightly informal word and carries an idea of a smaller or less important store. In the U.S. the word "store" is safer. You can always use it. You can use the word "store" for anything, big or small, and it sounds fine. You can call a gigantic Walmart a "store." A tiny store that sells soda, snacks, cigarettes and lottery tickets can be "a store" or "a shop." You would not call a Walmart store a "shop." You might say (U.S. usage) "the mall has two big anchor stores at each end, and fifty shops in between." "Shop" can be a verb. I can "go shopping" or "do the shopping." "Store" cannot be.

Shops and stores are synonyms. Shoe store sounds better.

There's definitely a difference, although it's not a terribly important one. A shop might not just sell goods -- it also often offers custom services. I would go to a "coffee shop" to order coffee a barista made for me at a cafe, and while I'm there I can buy some of the coffee beans they sell. I would go to a "coffee store" to buy a wide variety of coffee or appliances that make coffee, but I wouldn't expect a coffee STORE to have café services or places to sit and drink.