Are there any benefits to blowing everything up?

Yes, you can earn a "Destruction" experience bonus, a small amount of XP that varies with both level and number of objects destroyed.

The more you destroy, the larger the bonus. Since the bonus scales with level, it'll always be as (relatively) useful as you reach higher and higher levels.

Finally, there is an achievement for destroying 50 objects at once.

Beyond the Destruction bonus, they provide a way for a Barbarian to preserve and build Fury. The other classes with resource-generating attacks—Monk and Demon Hunter—gain no benefit from raging out like this as their pools either do nothing (Monk) or slowly recharge anyways (Demon Hunter).

You get bonus experience for going on destructive rampage. Destroy enough of the environment in a short period of time, and you will see that you earned a destruction bonus. The more environment you destroy during the streak, the higher the bonus experience.