Why did the people not destroy all the animals of the Amalekites?
(15,14) “And Shmuel said: What is this sound of sheep in my ears, and the sound of cattle that I hear? And Shaul said: They have brought them from the Amalekites; for the people spared the best of the sheep and of the oxen to sacrifice to Hashem, your G-d; and the rest we have utterly destroyed.”
We can explain this exchange according to the Midrash which asked: Granted the Amalekites were killed because they had sinned, but what was the animals' sin? It answered that the Amalekites were sorcerers and could change themselves into animals in order to save themselves. Therefore, Hashem commanded them to kill the animals as well.
Now, it says in Sefer Chasidim that those who change themselves into an animal are able to change their entire body except for their eyes, which remain the same. And in the gemora Bechoros it says that one of the blemishes which disqualify an animal as a sacrifice is if the eyes are similar to the eyes of a human.
Now we understand the exchange between Shmuel and Shaul: “Shmuel said: What is the sound of sheep which is in my ears” - you were supposed to also kill the sheep in case some of them are really humans. To this Shaul responded that “the people spared the best of the sheep to sacrifice to Hashem”. And since they took them only for the purpose of sacrificing, presumably they checked them for blemishes, including a check that their eyes are not similar to the eyes of humans. Therefore, none of them are Amalekites who changed themselves into animals, and so you have no cause to complain about our leaving these animals alive.
Why did Hashem reject Shaul from being king over Yisrael?
(15,26) “And Shmuel said to Shaul: I will not return with you, for you have rejected the word of Hashem, and Hashem has rejected you from being a king over Yisrael.”
We can explain how Hashem’s rejection of Shaul from being king was a direct result of his not killng the animals of the Amalekites according to the well known explanation of why the first king of Yisrael was chosen from the tribe of Binyamin, who was the youngest of the tribes. The answer that is given is that when Ya'akov met up with his brother Eisav in parshas Vayishlach, all the tribes bowed down to Eisav apart from Binyamin who had not yet been born.
According to this we can explain that Shmuel was saying to Shaul as follows: Hashem has rejected you from being king over Yisrael because you kept some of the animals of the Amalekites instead of destroying them. And even though you wish to justify your actions by saying that they were saved in order to sacrifice them to Hashem, you should still be guilty of the sin of 'maris ayin', because people might think that you took the animals for yourself. Therefore, the only way that your defence of your actions can be justified, is if you hold that 'maris ayin' is not a sin.
But the consequence of holding this opinion is that Hashem has rejected you from being king. Because why did Hashem choose you, the youngest of all the tribes, to be the king? Only because all the other brothers bowed down to Eisav. And even though the truth is that did not bow down to Eisav but rather to Hashem, they were nevertheless guilty of the sin of 'maris ayin'. But according to your opinion this is not a sin, and so the kingship was never fitting for you, but for Yehudah, "because Yehudah prevailed over his brothers, and the one appointed as prince was to be from him" (Divrei Hayomim 5:2).