How we understand these words depends on what part of the English-speaking world we come from. You could start by looking at this question where I, as a BrE speaker, asked how American English speakers used these two words:
A crude answer would be 'Shop is what Brits say and store is what Americans say' and another crude answer might be 'Stores are bigger than shops'. but the reality is more complex. Here's some history:
Traditionally, in the British Isles, people shopped in small, often family-owned businesses which sold only one category of product. Each town would have a High St with a range of little specialist shops, and customers would go to a baker for their bread, a butcher for their meat, a greengrocer for fruit and vegetables, and so on. For this reason, BrE speakers still tend to use the word 'shop', especially for smaller retail outlets. Today, a supermarket or retail chain might refer to their premises as 'a store', but ordinary customers tend to say 'shop'. If we hear or use the word 'store', this would suggest a fairly large outlet.
This is different from the US, where, in the past ( if I'm not mistaken) settlements and towns usually had a 'General Store' where people could buy full range of different products. This use of 'store' to mean 'a place where you can buy goods' is an extension of the original meaning 'a place where stocks of goods are kept'. Today (again, if I'm not mistaken), most people purchase all their goods from very large multi-category outlets such as Walmart and Target, referred to as 'stores'. For an AmE speaker, a 'store' can be a retail outlet of any size. See the answers in the link above for a few specific instances where AmE users might use 'shop' .