What was Saul's sin?
Essentially a form of rebellion against God.
Firstly the Israelites had rebelled against God's authority by demanding that they have a king - like the nations around them. God warned them through Samuel what having a king would be like, but the people insisted. So God let them have their wish and the consequences, by providing Saul as a king.
The job of a Godly king, is to mediate God's authority - much like the role of a priest as an intermediary between God and man. Saul however himself bowed to the pressures from his subjects and made decisions himself that might appear very pious, but were really rebelling against what God had asked him to do. For example, saving some of the animals of the Amalekites that he had been told to destroy and justifying it on the grounds that they were to provide a sacrifice.
It seems that Saul's heart was not towards serving God but looking good to the people.