If these two methods are simply synonyms, why do people go to the trouble of writing the additional characters \"_chain\"?
No.  alias_method is a standard method from Ruby.  alias_method_chain is a Rails add-on designed to simplify the common action of aliasing the old method to a new name and then aliasing a new method to the original name.  So, if for example you are creating a new version of the method method with the new feature new_feature, the following two code examples are equivalent:
alias_method :method_without_new_feature, :method
alias_method :method, :method_with_new_feature
and
alias_method_chain :method, :new_feature
Here is a hypothetical example:  suppose we had a Person class with a method rename.  All it does is take a string like "John Doe", split on the space, and assign parts to first_name and last_name.  For example:
person.rename("Steve Jones")
person.first_name  #=> Steve
person.last_name   #=> Jones
Now we're having a problem.  We keep getting new names that aren't capitalized properly.  So we can write a new method rename_with_capitalization and use alias_method_chain to resolve this:
class Person
  def rename_with_capitalization(name)
    rename_without_capitalization(name)
    self.first_name[0,1] = self.first_name[0,1].upcase
    self.last_name[0,1] = self.last_name[0,1].upcase
  end
  alias_method_chain :rename, :capitalization
end
Now, the old rename is called rename_without_capitalization, and rename_with_capitalization is rename.  For example:
person.rename("bob smith")
person.first_name  #=> Bob
person.last_name   #=> Smith
person.rename_without_capitalization("tom johnson")
person.first_name  #=> tom
person.last_name   #=> johnson
                                                                        alias_method_chain is worst way of doing method call interception. If you are looking for similar techniques, do not use it outside rails.