I think the very fact that Zuko was firstborn is partly why Ozai was bitter. Azulon seemed to favor Iroh (his firstborn) so Ozai took that out on Zuko. I think it gave him a sort of satisfaction, like he was avenging himself. Some sort of displacement.

I think that's part of the reason. He also seemed to be hard on Zuko because he wasn't a prodigy like Azula, and had to take more time in order to get better at firebending. But yeah, I think he was bitter towards Zuko for being the firstborn child.

^

Though mainly it was because he thought Zuko was weak. He saw him of having no potential, him being a cry baby and not a worthy heir.

What everyone else above said.

^^ And also because Ozai is a douche.

Not really. If their roles were switched, with Zuko being the second-born child and Azula being the eldest, Ozai would still treat them based on their power and use to him. We saw this during the time when Zuko returned from his banishment, Ozai treated him much more better and even arguably more favorably than Azula (like that war meeting when Zuko sat on his right and Azula sat on his left).