As a quick aside, sometimes it's not always because of some intrinsic benefit that things are one way or another.  They can also be byproducts of other adaptations, or [spandrels](http://en.wikipedia.org/wiki/Spandrel_(biology))

That being said, one example of a possible benefit is specialization.  For instance, birds will develop assyemtries in their visual system based on light inputs to their outward facing eye (one eye faces the egg shell, the other is obscured within the egg).  It was cause distinct assymetries in the brain that allow the bird to use one eyes and/or brain region for finding and tracking prey (a very robust detail-oriented object identifier) and the other will be turned towards the sky, looking out for predators (and isn't as detail oriented).  

![enter image description here][1]


  [1]: https://i.sstatic.net/ckyGC.jpg