The Maharal asks: Why did Avraham deny even as little as a thread or shoe strap from Sodom (14;23) and yet by Pharaoh he told Sarah to say she's his sister just so that he should get presents (Rashi, 12;13)!?!
Additionally, the Gemara states that the 4 Kings had the greatest army of all time. However, if so, how could Avraham put 318 people in grave danger by bringing them to fight at war (Rashi, 14;14)? We have a rule that we shouldn't put our lives in danger and rely on miracles to be saved (Kiddushin, 39b)!?!
Furthermore, Rashi (14;13) states that Og told Avraham that Lot was captured so that he would die at war and be able to marry Sarah. However, how did Og know that Avraham would put his life in danger for Lot? No one would go to Iraq and fight just because a family member was captured!?! Similarly, how could Og reasonably think that Avraham would put his life in danger to save Lot?
Rabbi Yochanan Zweig explains that Avraham Avinu sought to have a relationship with Hashem, unlike Noach. Avraham understood that there was a G-d who created everything and that the purpose of our creation is to give Him a place in this world. Therefore, Avraham tried to get everyone to have a relationship with Him. In contrast, according to the Zohar, Noach never even prayed for the people of his generation to repent.
Rabbi Zweig explains that the 4 kings were fighting against Avraham because he was following the will of Hashem and fulfilling his agenda. Therefore, Avraham was fighting for Hashem's sake. In the war against Pharaoh, however, he was fighting for himself. Therefore, he told Sarah to say she's his sister so that he would get gifts. Rabbi Zweig explains that there was no reason for Avraham to be scared of Pharaoh because he had already defeated millions of soldiers against Sodom. Therefore, explains Rabbi Zweig, Avraham didn't take anything from Sodom because it was Hashem's war and he wasn't fighting for his own personal benefit. Thus, Avraham didn't want to get benefit from a huge miracle and he rejected even a thread/shoe strap from Sodom.
Further, Rabbi Zweig explains that Og knew that the 4 kings captured Lot just so they could fight against Avraham as that was the main purpose of the war. Additionally, Og knew that Avraham wouldn't be scared to fight against the 4 kings because he had already beat Sodom, so the 4 kings were considered small in comparison. Thus, even though the 4 kings had a great army, Avraham wasn't putting 318 men in grave danger by bringing them to fight in the war.