Historically, the biggest difference between ginger beer and ginger ale is that ginger beer is brewed (fermented) and ginger ale is just carbonated water that has been flavored with ginger.
Ginger Beer origins date from the colonial spice trade with the Orient and the sugar-producing islands of the Caribbean.
Brewed Ginger Beer originated in Yorkshire in England in the mid-18th century and became popular throughout Britain, the United States, Ireland, South Africa and Canada, reaching a peak of popularity in the early 20th century.
However, modern day beers and ales may not be so clear cut. There are companies out there brewing ales and others spicing up carbonated water for their beers. The decision of which to choose comes from taste you prefer. Beers will typically be a much stronger ginger flavor than Ales.