If I remember correctly, it has something to do with the fact the Taepung and his family were poor, and his parents worked for Hosae's parents as driver and housekeeper / cook. Taepung felt embarrassed and resentful. The other kids at school may have also known about it which would make it more humiliating.
Yes, I think something about being teased at school came up, can't remember now if Hosae would defend him or not. It's unhealthy jealousy, not that any form of jealousy is 100% healthy lol, but Taepung's kind is dangerous. Resenting someone so much for what they have and you don't is always distasteful and a waste of time, especially if the person who's better off financially isn't stuck up and actually nice.
Now, I’m going only by the vaguest of memory, but I recall a brief segment where (I think) something was said about Taepung’s and Hosae’s school days, where there was some kind of trouble. Hosae’s rich parents took care of matters, but Taepung got thrown out of school for the same thing.
Thus (if I got it right) Taepung’s animosity towards Hosae isn’t so much personal, as much as it’s resentment of how his rich family uses money to get away with things that ordinary people will get punished for.
You're right - it was when Hosae was talking to his mother. He was telling her how much he disliked it when he and Taepung got into a fight and Hosae's parents used their money to keep Hosae in school while Taepung had to go elsewhere. Hosae felt that he was just as much at fault as TP and yet TP was the only one who got punished.