Let's assume that you did blow up an asteroid, and the constituent parts loosely reassembled. Wouldn't they they break apart again as soon as they started to experience the turbulent effects of the Earth's atmosphere? And once they did, the surface area to mass ratio of the asteroid matter would be an order of magnitude or higher, pretty much guaranteeing all of the mini-asteroid parts would burn up.
No doubt I'm wrong. I'm a programmer, not a physicist or an astronomer. So I'm seeking to get 'educated' more than I am to argue with you.
On top of that, nudging it out of a collision trajectory is going to require a lot less energy than disintegrating it, particularly if you can arrange to do the nudge a long way off. What's easier, pushing a runaway shopping cart so it doesn't bang into your car, or tearing the cart into parts small enough that they don't do any damage when they hit?
Neal Stephenson's latest novel, Seveneves has some discussion of this (on a much larger scale, granted).