A part of Wilbur's soul is missing.
Now hear me out. When Wilbur was revived he switched places with Ghostbur. They didn't become one of the same entity, they switched places.
Why is this important you may ask?
If you remember, when Ghostboo was talking to aimsey he told her that he's half of himself, the part that stayed. While his counterpart went through hell. Ghostboo explained that he was the part that held the memories, while the one in limbo was feeling everything. All the bad things happening in Limbo and what happens in the dsmp. What's important about this is that the one in Limbo feels and the one on the dsmp is responsible for memories.
Ghostbur was the one who feels, and the Wilbur in Limbo was responsible for all the memories. (Its also why when Wilbur was revived he couldn't quite remember Ghostbur's memories, because Ghostbur wasn't a reliable source). And what's sad about their story is that they didn't stay in touch like Ranboo did. Ranboo found a way to communicate with both parts of himself and find a way to keep his emotional self stable until they were revived. If one part of him fades the soul can not be put back together... Meaning if the emotional or mental part of the soul gets overloaded it fades. This is exactly what happened to Wilbur. Ghostbur and Revivebur are one in the same just different parts of one soul. Both of them hated each other, if you look back on them talking about each other you can tell neither of them were quite fond of the other. (His emotional soul hated the mental soul for hurting the things and people he loved. The mental part of the soul hated the emotional part for... feeling). So essentially Ghostbur and Deadbur gave up on each other. (Or Ghostbur gave up on Deadbur which would be pretty dark and sad). So yeah, Dream HAD to kill ghostbur because both of them couldn't be revived. Of course it's clear that the part of Revivebur's soul that feels didn't completely dissipate. Its just clear that its not the same. Dream revived Wilbur's mind, and killed Wilbur's emotions. (A good chunk of it atleast)