Big and large are often found to be used as synonyms. That means that they have very similar meanings. However, there is a difference between them. Large is often used to describe a size whereas no one would use the word “big” to describe a size whether it was clothing or the size of a drink. It does seem that there is a difference in the connotation of it.
Large has a slightly more positive connotation than big. Big has a slightly negative connotation because it would imply that something is overweight. Whenever you want to order something that comes in size, the person will ask you if you want small, medium or large. You never hear someone ask if you want something in small, medium or big.