How has God differentiated the people who are called His children of promise and those whom He calls the children of God?
The Bible uses the term “child of promise” to refer specifically to Isaac, the son God promised to deliver to Abraham and Sarah. The promise God gave to Abraham included the provision that God would bless all nations through Abraham’s Seed (Jesus), and those who receive that blessing by faith in Jesus are known as the spiritual children of that promise.
Therefore, the New Testament uses the term “children of promise” to refer to all who are born again by faith in Jesus Christ, the Seed promised to Abraham. Therefore, all who are children of the promise are also those who have become children of God by faith in Jesus Christ.