What is the difference between 例子 and 榜样?
榜样 is a model, specifically a person regarded as an excellent example of a certain quality.

Every year, the government announces


  全国劳动模范 or 劳模 in short (people who excel in work) and 全国道德模范 (people who possess good
  moral values)


In a lot of companies, they have an honor called


  模范带头人 (employees that come first in sales numbers, etc.)


Teachers use the word fairly often:


  小明是我们（学习）的榜样。Xiao Ming set a good example for us.


An idiom:


  榜样的力量是无穷的。If one sheep leaps over the ditch, all the rest will
  follow. (official translation)




例子 is a typical thing that can represent the whole kind.


  举个例子 give an example




It's actually very easy to distinguish between the two words.

Definition of "example" (from the Dashboard widget on my Mac):


  example |igˈzampəl| noun
  
  
  a thing characteristic of its kind or illustrating a general rule --> 例子 (neutral)
  a person regarded in terms of their fitness to be imitated or the likelihood of their being imitated --> 榜样 (positive, and used to describe people
  only)