Saul and David

When did God decide to remove Saul, Israel's first king, and raise up David in his place?
When Saul sinned by making a sacrifice to God that wasn't his to make: Then the Philistines gathered together to fight with Israel, thirty thousand chariots and six thousand horsemen, and people as the sand which is on the seashore in multitude. And they came up and encamped in Michmash, to the east of Beth Aven. When the men of Israel saw that they were in danger (for the people were distressed), then the people hid in caves, in thickets, in rocks, in holes, and in pits. And some of the Hebrews crossed over the Jordan to the land of Gad and Gilead. As for Saul, he was still in Gilgal, and all the people followed him trembling. Then he waited seven days, according to the time set by Samuel. But Samuel did not come to Gilgal; and the people were scattered from him. So Saul said, “Bring a burnt offering and peace offerings here to me.” And he offered the burnt offering. Now it happened, as soon as he had finished presenting the burnt offering, that Samuel came; and Saul went out to meet him, that he might greet him. And Samuel said, “What have you done?” Saul said, “When I saw that the people were scattered from me, and that you did not come within the days appointed, and that the Philistines gathered together at Michmash, then I said, ‘The Philistines will now come down on me at Gilgal, and I have not made supplication to the LORD.’ Therefore I felt compelled, and offered a burnt offering.” And Samuel said to Saul, “You have done foolishly. You have not kept the commandment of the LORD your God, which He commanded you. For now the LORD would have established your kingdom over Israel forever. But now your kingdom shall not continue. The LORD has sought for Himself a man after His own heart, and the LORD has commanded him to be commander over His people, because you have not kept what the LORD commanded you.” (1 Samuel 13:5-14)

But didn't David sin as well?
Yes, he committed adultery against one of his most loyal soldiers, and then murder when his efforts to cover up the adultery failed: Then it happened one evening that David arose from his bed and walked on the roof of the king’s house. And from the roof he saw a woman bathing, and the woman was very beautiful to behold. So David sent and inquired about the woman. And someone said, “Is this not Bathsheba, the daughter of Eliam, the wife of Uriah the Hittite?” Then David sent messengers, and took her; and she came to him, and he lay with her, for she was cleansed from her impurity; and she returned to her house. And the woman conceived; so she sent and told David, and said, “I am with child.” Then David sent to Joab, saying, “Send me Uriah the Hittite.” And Joab sent Uriah to David. When Uriah had come to him, David asked how Joab was doing, and how the people were doing, and how the war prospered. And David said to Uriah, “Go down to your house and wash your feet.” So Uriah departed from the king’s house, and a gift of food from the king followed him. But Uriah slept at the door of the king’s house with all the servants of his lord, and did not go down to his house. So when they told David, saying, “Uriah did not go down to his house,” David said to Uriah, “Did you not come from a journey? Why did you not go down to your house?” And Uriah said to David, “The ark and Israel and Judah are dwelling in tents, and my lord Joab and the servants of my lord are encamped in the open fields. Shall I then go to my house to eat and drink, and to lie with my wife? As you live, and as your soul lives, I will not do this thing.” Then David said to Uriah, “Wait here today also, and tomorrow I will let you depart.” So Uriah remained in Jerusalem that day and the next. Now when David called him, he ate and drank before him; and he made him drunk. And at evening he went out to lie on his bed with the servants of his lord, but he did not go down to his house. In the morning it happened that David wrote a letter to Joab and sent it by the hand of Uriah. And he wrote in the letter, saying, “Set Uriah in the forefront of the hottest battle, and retreat from him, that he may be struck down and die.” So it was, while Joab besieged the city, that he assigned Uriah to a place where he knew there were valiant men. Then the men of the city came out and fought with Joab. And some of the people of the servants of David fell; and Uriah the Hittite died also. (2 Samuel 11:2-17)

Then why did David but not Saul find favor with God?
Their difference wasn't in their sin, since both sinned and it can be argued that David's betrayal, adultery and murder were more grave sins than Saul's impatient sacrifice. Their difference lay in what they did afterwards when confronted about their sins. Saul made excuses and blamed others as above, while David admitted his sinfulness: Then the LORD sent Nathan to David. And he came to him, and said to him: “There were two men in one city, one rich and the other poor. The rich man had exceedingly many flocks and herds. But the poor man had nothing, except one little ewe lamb which he had bought and nourished; and it grew up together with him and with his children. It ate of his own food and drank from his own cup and lay in his bosom; and it was like a daughter to him. And a traveler came to the rich man, who refused to take from his own flock and from his own herd to prepare one for the wayfaring man who had come to him; but he took the poor man’s lamb and prepared it for the man who had come to him.” So David’s anger was greatly aroused against the man, and he said to Nathan, “As the LORD lives, the man who has done this shall surely die! And he shall restore fourfold for the lamb, because he did this thing and because he had no pity.” Then Nathan said to David, “You are the man! Thus says the LORD God of Israel: ‘I anointed you king over Israel, and I delivered you from the hand of Saul. I gave you your master’s house and your master’s wives into your keeping, and gave you the house of Israel and Judah. And if that had been too little, I also would have given you much more! Why have you despised the commandment of the LORD, to do evil in His sight? You have killed Uriah the Hittite with the sword; you have taken his wife to be your wife, and have killed him with the sword of the people of Ammon. Now therefore, the sword shall never depart from your house, because you have despised Me, and have taken the wife of Uriah the Hittite to be your wife.’ Thus says the LORD: ‘Behold, I will raise up adversity against you from your own house; and I will take your wives before your eyes and give them to your neighbor, and he shall lie with your wives in the sight of this sun. For you did it secretly, but I will do this thing before all Israel, before the sun.’” So David said to Nathan, “I have sinned against the LORD.” (2 Samuel 12:1-13)

Does this distinction apply today?
Yes, because we are imperfect, fleshly human beings, Christians still sin from time to time despite our best efforts to not sin: "For all have sinned and fall short of the glory of God" (Romans 3:23). The key questions are, 'What do we do thereafter? Do we make excuses and try to justify ourselves, or do we admit our sins, come before the Lord in repentance and ask Him to forgive us?' The difference in outcome that hinges on these questions is no less than that between Saul and David, or Judas Iscariot and Peter.