In Shmuel Alef chapter 15 Shaul HaMelech (King Saul) loses his kingship because in his battle against Amalek he failed to kill King Agag and saved livestock to offer as sacrifices.
Later we find that not only did Dovid keep plunder after battling Amalek (Shmuel Alef chapter 30) he institutes regulations about how to divide plunder.
Why is Dovid [apparently] not censured while Shaul lost his throne?
What's right or wrong is dependent on the will of God, which may differ in differing situations.
King Saul had clearly been (by Samuel) told to kill all the animals, and he disobeyed.
King David was given no such order; in fact, the Urim V'Tumim told him "go save!" (hatzel); the same language used in v.18, he "saved" that which had been taken.
As to presumably why the orders were such, see Alex's answer.
From vv. 18-19, it seems that David only took back what the Amalekites had previously plundered (and per verse 16, that was from the Jews and, lehavdil, from the Philistines):
יח וַיַּצֵּל דָּוִד, אֵת כָּל-אֲשֶׁר לָקְחוּ עֲמָלֵק; וְאֶת-שְׁתֵּי נָשָׁיו, הִצִּיל דָּוִד. יט וְלֹא נֶעְדַּר-לָהֶם מִן-הַקָּטֹן וְעַד-הַגָּדוֹל וְעַד-בָּנִים וּבָנוֹת, וּמִשָּׁלָל, וְעַד כָּל-אֲשֶׁר לָקְחוּ, לָהֶם: הַכֹּל, הֵשִׁיב דָּוִד.
The Gemara (Bava Kamma 114a) states that if a Jew rescues another Jew's property from a non-Jewish bandit, we have to assume that the original owner didn't give up hope of retrieving it, and therefore it has to be returned to its owner. This is cited as halachah in Shulchan Aruch, Choshen Mishpat 368:1. So the Amalekites never actually legally acquired whatever they had taken from Tziklag and other Jewish towns, and thus David was fully justified in taking what was originally his and his men's (and returning whatever was recognizable to its original owners). Whereas in Shaul's case: even if we grant that perhaps some of the Amalekite property was also previously plundered from the Jews or from other nations, it wasn't recognizable as such.
Not only does Shaul not follow the very clear orders of Shmuel in this instance he has also already been warned about not retaining Hashem's help in his kingship when he made a sacrifice prior to a war instead of waiting for Shmuel. He is therefore more guilty than David, who has not been warned AND was not given direct instruction.
