Saul was chosen by the Lord to be the first King in Israel. He was admired by the people and had many talents and was very tall. He was anointed to be captain over Israel, though he was later called King. The anointing was a symbol of the endowment of the spirit from God and the monarchy became a Divine Institution to help fulfill the purposes of God.
This is where I believe Saul's jealousy began. It began with shame and guilt for doing something against the Lord's will. The jealousy of David became the manifestation for this. Jealousy tends to turn focus from changing and improving oneself to blaming others. Saul could never overcome this. He tried, he even swore not to kill David. But he could only focus on things that he didn't have instead of things that he did have (which is how Heavenly Father sees us according to Joseph Smith).
President Ezra Taft Benson gave a life-changing talk "Beware of Pride" he said that with prideful people, the praise of men rings more loudly than the praise of God. I have thought a lot about this. It is obvious what the praise of men is, but how do we recognize the praise of God for us and what do we do with that recognition. It almost seems incorrect to say that God has praise for me, but I really believe that he does. Not only does he have praise for us, but recognizing that praise is essential in our well-being, image and peace. It is not something that we have to report to anyone, in fact we should probably be like Mary and ponder it in our hearts, just letting that confirmation rest between me and the Lord. But if we can figure out how to recognize praise from the Lord, I think we can avoid the jealousy trap that Saul was in.