It was a very basic reason called being true to your salt, do not betray your loyalty. Drona's early life was one of poverty, he and his family had to suffer a lot because of it. The incident when his son Aswathamma, was fooled into believing a mixture of rice powder and water as milk, hurt him deeply, and he resolved, his family would never be in poverty again. He approached his childhood friend Drupada, to help him out of his misery. However Drupada, in his arrogance, insulted and rebuked him, and sent him away. He walked out swearing to have his revenge.
Fortunately his brother in law, Kripacharya,( brother of his wife Kripi), bought him to Hastinapur. There is that anecdote about how Drona pulled out a ball from a well, using only blades of grass tied together, and this impressed Bhishma. Anyway knowing well of Drona's prowess with Shastra Vidya, Bhishma appointed him as the tutor for both the Pandavas and Kauravas. Yes Arjun was his favorite student, but that was strictly more of a teacher-student relationship, an admiration for Arjun's skills as an archer. While Arjun was his favorite student, and he admired Yudhistir for his truthtfulness, he could never betray his loyalty to Hastinapur, which had given him everything.
When it came to the final war, Drona went along with the Kauravas, because there was no way he could betray Hastinapura. It was not just the Pandavas, even if his own son Aswathama, had joined the other side, he would have still fought against him. Another factor of course was that his life long rival, Drupada, was on the Pandava's side. And there was no way he could ever have fought along with him on the same side.
The realtion of Kripacharya with Drona led him to fight against Pandavas.
They Fought because they were being true to their salt and not paying for the grace is not the duty of a real Brahman.In fact, if the brahman can't pay for the salt how can he/she follow other brahman duties like pray for the society and secure the blessings of the gods.
Apastamba say's so not counting the boundation of the situation but manusmrita adds further.
Whenever you read various scriptures describing same topics relate both of them if it relates no doubt but if it doesn't put a question on it.The two scriptures you mentioned in the question are not opposing each other instead the later describes it more and the former doesn't so because if even a single topic is described with atmost elaboration the millions of pages will be required.