I don't think I understand your question. The words can be used interchangeably in some situations, like "Put 2 and 3 together" or "Add 2 and 3 together." However, I would say, "Put the book on the shelf." I wouldn't say "Add the book on the shelf."
To put is transfer something or place it on or in a place or container that does not have another thing of the same kind. To add is to place in addition to something of same kind increase or top up. This is only in one aspect of the meaning.