Possible Benefits:

Simply put, when we train our body adapts.   That is how we get bigger and stronger.  However, if we train the same way for more than a few weeks at a time, our body will stop responding to the training, and we need to do something different to get our body to respond again.  This is the driving force behind perhaps the most popular training ideology in the world (particularly the Western Hemisphere) since it first became popularized by Tudor Bompa in the 1960's - periodization.  If you are in competitive athletics, this is more than likely the type of system that your coaches have you using. 

Until recently, the only ways to help create the variation needed for continued adaptation were to: 1) change how the exercise is done by modifying the movement somehow, 2) change the set/rep scheme, 3) change the amount of weight being used, 4) change exercises all-together.  Resistance bands provide another means to create variation to get the necessary adaptation.

In addition, they may be more effective in improving power (explosiveness) and force (strength) than traditional weight training.  These details are covered in the "Research" section.