King Shaul’s error
ויבא שמואל אל-שאול ויאמר לו שאול ברוך אתה ליקוק הקימתי את-דבר יקוק: ויאמר שמואל ומה קול-הצאן הזה באזני וקול הבקר אשר אנכי שמע וגו’ הלוא אם-קטן אתה בעיניך ראש שבטי ישראל אתה וגו’
Shmuel came to Shaul, and Shaul said to him: “You are blessed to Hashem! I have fulfilled the word of Hashem”. Shmuel said: “Then what is this sound of the sheep that is in my ears? And the sound of the cattle which I hear? … You may be small in your eyes, but you are the head of the tribes of Israel!…
The haftarah for parshas Zachor details the failure of King Shaul to eradicate the wicked nation of Amalek. Shmuel the prophet ordered Shaul to leave no person or animal alive, as Hashem told Moshe that we are to blot out the memory of Amalek. Shaul however left alive the king of Amalek known as Agag, the ancestor to Haman. He also left alive their sheep and cows, intending to bring them as offerings for Hashem. Shmuel harshly reprimanded Shaul for his failure, and Shaul lost the kingship as a result of his sin.
Now, Shaul was commanded to leave no remnant of Amalek. He knowingly went against these instructions. What was the thinking? Furthermore, when he met up with Shmuel, he greeted him by exclaiming that he had fulfilled the word of Hashem. How could Shaul say something that was patently false? Shmuel immediately points out that he hears the animals of Amalek still alive. Shmuel then says something strange. He says that Shaul thinks of himself as small, and this is mistaken. What does that have to do with anything?
The commentaries provide two rationales for why Shaul was commanded to not only destroy Amalek, but their animals as well. One explanation is that the Amalekite people were proficient in sorcery, and were able to shapeshift into animals. If the animals were to be then left alive, some of them might have been really Amalekites in disguise. However, this explanation creates a potential chillul Hashem, desecration of Hashem’s name. It almost implies that Hashem is afraid of their sorcery. We are taught that Rabbi Chanina ben Dosa was impervious to witchcraft, as he was a righteous individual. For sure the entire Jewish people, who had the Divine Presence amongst them, had no reason to fear Amalek’s magic.
Another explanation that is suggested is simply that we are to erase any memory of Amalek. If their animals would remain, people would be reminded of this nation. However, according to this reason, it would have been permissible to bring their animals as offerings. At the end of the day, the animals would be gone, and along with them any memory of Amalek. In contrast, according to the first reasoning, this wouldn’t have been an option. The animals needed to be removed alongside the Amalek people themselves. Any delay could result in an Amalekite disguised as an animal to escape.
Shaul couldn’t fathom that the reason for Shmuel’s command could be a concern for sorcery. Such fears could lead to a chillul Hashem. He therefore understood it to be the second reason, and thus felt justified in keeping the animals alive for future offerings. He also kept some of the animals alive so no one would think the command was because of the first reason. By keeping the animals alive, the people would understand that the intent was to remove any memory of Amalek. This is what Shaul was thinking when he told Shmuel that he fulfilled the word of Hashem. By keeping the animals alive, he simultaneously avoided a chillul Hashem, gathered offerings for the Temple, and will still succeed in erasing the memory of Amalek.
However, he made a grave error. We can’t ever fully understand the reasonings behind the mitzvos. At the end of the day, they are decrees from Hashem. Since the prophet told him to wipe everything out, Shaul should have done so. Shmuel brought Shaul a proof that this was what he should have done. He pointed out that Shaul considered himself small in his eyes. This was the case, despite the fact that he was the king of Israel. Let’s assume that Shaul was indeed a person of small spiritual stature. We see nevertheless that he is king. It must be that Hashem knows better, for He chose Shaul to be the king. Although this didn’t make sense to Shaul, he had to admit that it was a decree from Hashem. The mitzvah regarding Amalek was no different.
 Based on Chasam Sofer’s Toras Moshe III L’Parshas Zachor s.v. § יש להתבונן על שאול
 I Samuel 15:13,14,17
 Exodus 17:14
 Megillah 13a
 Rashi to I Samuel 15:3; Ba’al HaTurim to Exodus 22:17 in the name of an unknown Midrash, brought by Chanukas HaTorah Likkutim Haftarah L’Parshas Zachor
 Chullin 7b
 Rashi to Deuteronomy 25:18, quoting Midrash Lekach Tov ad. loc. Rashi seems to contradict himself, as he also said the reason was because of sorcery. I found this addressed in Shoshanas Yisrael parshas Mordechai haftaras parshas Zachor, by Rav Mordechai Kanner. He suggests an answer based on the Oneg Yom Tov Hakdama § 6 in a footnote, who holds the command to destroy Amalek’s animals only applies after the people of Amalek are annihilated. Only then would it make sense to destroy “their memory”, once they are gone yet their animals are alive. If so, Shaul was correct in leaving the animals, as Agag was still alive. This would be true in general. However, as Rashi points out in that story, there was a special command to destroy the animals at that time, due to a concern for sorcery. Therefore, Shaul was reprimanded for not dealing with the animals right away
 Similar to a בהמה הנרבעת (Sanhedrin 7:4 and 54a, brought by Rashi to Leviticus 20:15). The Oneg Yom Tov loc. cit. also says this
 The opinion of the Rashba is that delving into the topic of the reasoning behind mitzvos is a futile process
 Berachos 33b; Megillah 25a