Why did God's mercy remain on David's line but not on Saul?

God's mercy remained on David's line because of His promise that David's royal house, kingdom, and throne would be established forever, ultimately bringing the King of kings. Though David's descendants would be chastened for iniquity, God would not remove His mercy as He did from Saul, whom He put away.

Les teaches that in 2 Samuel 7 David is promised that from his loins would come a royal house or family of kings to rule and reign over Israel, and through that line would one day come the King of kings—Christ, born of Mary with Joseph as His legal father, both descending from the royal bloodlines of David and Solomon. God tells David that if his line commits iniquity, He will chasten them with the rod of men, but His mercy will not depart from him as He took it from Saul, whom He put away. Then God promises David's house, kingdom, and throne will be established forever. Les stresses God cannot lie, and this royal family goes beyond history into eternity; out of David's royal family would come the King of kings and Lord of lords, who will come again to the nation of Israel.

2 Samuel 7:15

But my mercy shall not depart away from him, as I took it from Saul, whom I put away before thee.

What does the 'But' at the start of II Samuel 7:15 teach about God's mercy toward Israel?

The 'But' in II Samuel 7:15 contrasts Israel's failures and God's discipline with His unwavering mercy. Even when the nation falls short or becomes unbelieving, God declares that His mercy shall not depart from them. Though He chastens them repeatedly with the rod of men, they remain under His thumb.