a==b is used to compare references where as a.Equals(b) is used to compare the values they are having.

For Ex:

 public class Class1
    {
        string name = String.Empty;
        public Class1(string strName)
        {
            strName  = name;
        }
    }

Class1 a = new Class1("Hello");
Class1 b = new Class1("Hello");
a==b // Returns false
a.Equals(b) // Returns true