Their meaning is a mostly the same. "Little" is a bit more casual/conversational/spoken, while "small" is more formal or written. "Please go pick up the little box in the corner of the room." "Our product arrives in a small, well-designed box."

Small is used to talk about the size of something. Example: Your cat is very small. Can I have two small pizzas please? Little can be used in the same way as small to refer to size, but it can also be used with another adjective to express an emotion. Example: He’s a clever little boy. They also collocate differently and can convey slightly different meaning. Example: You might be angry at a person for behaving immaturely, belligerently or inappropriately and call them " a pathetic little man" 'Little' works here whereas 'small' does not. Note the insult in the name of the character 'Littlefinger' in Game of thrones for example. 'Smallfinger' just wouldn't have sounded quite as insulting! Nobody’s looking after that poor little dog.

Small box. "A little" box would mean that the box is very dear to you, you like it a lot despite it being small.