When he was a child, Dante admired Simon for always knowing the answer to everything. When asked, he would never hesitate to give a proper answer, while Dante always struggled with everything. He hid that admiration by constantly bullying Simon, trying to get him to pay some attention to himself, even though it never worked. Through time, that admiration became respect and finally strong desire to surpass him.
While Dante stayed true to the teachings of his mentor, Simon began to stray from them, not being able to handle the trauma he experienced as a child. He became more ruthless as time passes with his actions, which involved mass manslaughter, practice and search of dark magic as well as wanting to revive Zeref's power and use it to purify the world.
While knowing that he could have killed him long ago, but didn't, Dante partly blames himself and hates Simon for what he has become. He has sworn, even at the cost of his very soul, that he would find him and kill him with his own bare hands.