Amber Ale is a term used to describe pale ale beers that are brewed using amber and crystal malts, which give them their characteristic amber color they are known for. They typically have a smooth finish and a less dominant hop profile (although there are exceptions). They're a great every-day beer that can be enjoyed at anytime and any season.
Brown Ales are a style of beer that get their name from it's use of dark and brown malts, which impart the characteristic dark amber and brown color they are known for. They originate from 17th century London, when brewers would use the term to more easily convey their beers to their customers.