Well, there's a very detailed breakdown you can read about here: (x)
Really, the biggest difference is Henry being turned into a comic relief character, and the sort of impression he gives off due to this change.
Ehh, I wouldn't call him genuinely insane. He only tried to kill the dog in order to prevent it from suffering prolonged pain before dying. A quick death was better than eventual starvation (due to the broken leg), in his mind. After Olivia says she'll heal it, Henry immediately offers to help. He says, roughly:
"I thought that [death] was its natural fate.
But if it can get help, it's not its time yet.
If it can come back from the brink of death,
nothing could be better~!
Isn't that right, Barky?
It's good to be saved~"
source: (x)