Why would you use String.Equals over ==?

Both methods do the same functionally - they compare values.
As is written on MSDN:

• About String.Equals method - Determines whether this instance and another specified String object have the same value. (http://msdn.microsoft.com/en-us/library/858x0yyx.aspx)
• About == - Although string is a reference type, the equality operators (== and !=) are defined to compare the values of string objects, not references. This makes testing for string equality more intuitive. (http://msdn.microsoft.com/en-en/library/362314fe.aspx)

But if one of your string instances is null, these methods are working differently:

string x = null;
string y = "qq";
if (x == y) // returns false
    MessageBox.Show("true");
else
    MessageBox.Show("false");

if (x.Equals(y)) // returns System.NullReferenceException: Object reference not set to an instance of an object. - because x is null !!!
    MessageBox.Show("true");
else
    MessageBox.Show("false");