Question:What is factory method? 

Using factory method, we first define a factory and then assign method to it.
var mainApp = angular.module("mainApp", []);
mainApp.factory('MathService', function() {     
   var factory = {};  
		
   factory.multiply = function(a, b) {
      return a * b 
   }
   return factory;
});

Question:What are the differences between service and factory methods? 

factory method is used to define a factory which can later be used to create services as and when required whereas service method is used to create a service whose purpose is to do some defined task.