What is the problem with blowing up large asteroids?

Often on television shows and in articles I see it mentioned that it'd always be bad to blow up an asteroid or comet because then the energy would just be spread out and cause even more damage.

According to some estimates I've seen around 100 tons (or more) of meteoroids hit the Earth each day. If all of this were combined together into a single asteroid, it could destroy an entire, large city.

Given that alone, it makes it seem to me that it be more logical to take the chance and blow up an asteroid and thereby trim its weight down, causing a lot more to be more easily burnt up on entry, so that when it did hit, it would cause less damage.

Is this logical at all? If my science/math/physics is incorrect, I want to understand why having more burn up by spreading out is worse compared to having it more concentrated and vastly more dangerous.