Fathers and sons fight. It could be something in our DNA that makes us as men get primal and fight for superiority against our dads. In some mythologies, it's a right of passage for a father and son to fight to the death because only one of them can survive. Kronos and Zeus is a great example of father and son hatred. In comic books, Wolverine and Daken may have taken the sons and fathers fighting to a whole new level.
Daken hates his father. It doesn't matter that he was fed a lie that Logan left him and his mother for dead and the guy who raised him was the one who set things in motion for Daken's mother's death. All Daken knows is that Logan was never there for him and he was left to be raised by a monster of a person who turned him into a killer when he could have had a much different life. And there was nothing Logan could do would make up for it. Daken even admitted that a lot of his hatred stemmed from jealousy. Logan would became a teacher and father to other kids but not for him. This hatred lead Daken to go above and beyond the normal and ended up with Daken's death.
The age old phrase "This town ain't big enough for the two enough" and Daken felt like him and his father could not coexist. So Daken joined up with Wolverine's most hated enemy Sabretooth. Daken captured one of Logan's students from his school and forced Logan's hand. Daken trapped Logan and would lament about how Logan was never there for him, but Logan still didn't hate his son no matter what he did. In fact, Logan said he felt bad for him. Saying Daken was desperate to show his father how good he was. Logan pleaded, but Daken was set on killing him and almost succeeded. But Logan was warned by his future self (It's a comic book. It's supposed to be wacky) that if he didn't kill Daken, that Daken would kill the students at his school. So Logan did what had to be done. Logan killed his son to protect the school and everyone in it.
Later on, Daken does come back and does try and get revenge on his father and does. But, when Logan got his chance to get back at his son, Logan takes him out like he always does. What's funny about this rivalry was after Logan died, Daken was actually sad. Maybe he actually cared about his father regardless of what they went through. Or maybe he was just upset that he wasn't the one to kill him. Judging by the way he took his death, he really did care about his father.