Title is misleading. Turning an asteroid into a cloud of debris with an intact core would not be enough to save the earth because the debris would reform around the core of the asteroid.

That's not the same as being unable to blow up an asteroid to save the planet - you just need a more energy to also destroy the core, or followup to scatter debris and overcome gravity, etc.

At the energy levels required it's probably easier to just move the asteroid off of a collision course, so even an advanced enough civilization to blow up an asteroid probably wouldn't. They shift its course, or trap it and mine it, etc. But that isn't the same thing as never being able to do it. There's nothing special about asteroids that make them impervious to explosions.