There’s actually two surviving versions of the David and Goliath story. One is represented by the Greek Septuagint (LXX); the other is represented by the Masoretic Text and the Vulgate. The version represented by the Greek is the shorter, smoother one. The longer Hebrew version, on the other hand, has a number of continuity errors that give it away as a possible combination of two different versions of the David-Goliath story; the MT has combined the LXX source and (at least) one other source, grafting the latter to the former, but not always smoothly.
In the shorter version:
David is under Saul’s service as his armor-bearer and harpist (16:15-23). During an Israelite-Philistine battle, Goliath - measuring an impressive, but not totally unusual “four cubits and a span” (six feet nine inches) - appears to challenge the Israelites in the hearing of Saul and David (17:1-11). David volunteers to fight Goliath, recounting how he used to kill lions and bears with his bare hands, back when he was a shepherd boy. He manages to kill Goliath using his slingshot (17:32-49). Beheading Goliath with a sword, David takes the head to Jerusalem - which would have still been under Jebusite domination then (17:51-54). Saul grows increasingly jealous of David when the people start praising David more than him (18:6-9), so he removes David from his direct service and appoints him “a commander of a thousand;” David’s increasing popularity frightens Saul even more (18:12-16). He then tries to offer David his daughter Michal in marriage in exchange for 100 Philistine foreskins (secretly hoping to get rid of David); David manages to complete the task - to Saul’s bewilderment - and wins Michal (18:20-28).
That’s how the shorter LXX version goes. The longer version (MT / Vulgate) combines this shorter story with extra material, not always totally harmonized smoothly with the short story. In addition, there are little alterations in the shorter version as well.
(17:4) Goliath is now clearly a giant, measuring “six cubits and a span” - i.e. over 9 feet tall.
(17:12-31) David is portrayed as still being an unknown, young shepherd boy rather than Saul’s armor-bearer. It is his three of his eight brothers, not him, who serve in Saul’s army. David’s presence in the battlefield is explained as being due to him delivering food for his brothers. David overhears the rewards promised to the one who can defeat Goliath (“The king will enrich the man who kills him with great riches and will give him his daughter and make his father’s house free in Israel”). The eldest of the eight brothers, Eliab, chides David for coming to the battle: “I know your presumption and the evil of your heart, for you have come down to see the battle.” David brushes off his rebuke - he was just asking (“Was it not but a word?”). David’s words were overheard and is then brought before Saul.
(17:43) David’s biting remark to Goliath’s taunt found in the Greek version (“Am I a dog, that you come to me with a stick and stones (Hebrew ‘sticks’)?” - “No, but worse than a dog” ;)) is omitted in the Hebrew.
(17:50) The Hebrew adds this phrase: “So David prevailed over the Philistine with a sling and with a stone, and struck the Philistine and killed him. There was no sword in the hand of David.” In the shorter version, it’s unclear whose sword it was that David used to kill Goliath; the natural reading would be to be infer that it was David’s own sword (“And David ran and stood over him and took his sword and put him to death and cut off his head”). But in the longer version, the sword is strongly implied to be Goliath’s ("There was no sword in the hand of David. And David ran and stood over the Philistine and took his sword and drew it out of its sheath and killed him and cut off his head with it.
(17:55-58) Saul clearly does not recognize David the young shepherd boy; he has to ask his commander Abner who he is. Goliath’s head is said to be still “in [David’s] hand.” (What happened to David’s sending it to Jerusalem?) David introduces himself to Saul as though the man and boy had never met.
(18:1-6a) The covenant between David and Jonathan (vv. 1-5a) is not in the Greek. In the short version Jonathan is only to be introduced later in chapters 19-20, where he (as in the longer version) is sympathetic to David against his father Saul. Saul in v. 5 is said to have “set [David] over the men of war” (contradicting ch. 16, where Saul had already appointed David a position in the army as his armor-bearer).
Whereas in the shorter version, the Israelite women come out to meet and praise David, clearly in the context of his victory over Goliath (18:6-9 immediately follows chapter 17), in the longer version, the women are said to come out to meet Saul, but their song does not so much praise him as David instead.
(18:8-12) The longer version adds the incident about Saul being tormented by “a harmful spirit” and throwing a spear at David (a version of which is also found in 19:9-10). The shorter version has a more natural sequence of events: Saul is first envious (18:8-9), then suspicious (18:12), then frightened of David’s success (18:13-15). Saul wants David killed by the Philistines (18:21), then by his own sons and servants (19:1), and when all that fails, he tries to kill David himself (19:9-10). The longer version kinda ruins this flow by prematurely adding the story of Saul’s failed attempt to skewer David to the wall.
(18:17-19) The longer version adds Saul offering his eldest daughter Merab to David first (as per custom), which he turns down. At which Saul then offers Michal.
(18:27) The shorter version has David successfully collecting one hundred Philistine foreskins for Saul. The longer Hebrew version ups the ante by having David collect two hundred foreskins - twice that of Saul’s demand.