.net – Is the .NET Framework installed by default on Vista

string is an alias in C# for System.String.
So technically, there is no difference. It's like int vs. System.Int32.

As far as guidelines, it's generally recommended to use string any time you're referring to an object.

e.g.

string place = "world";

Likewise, I think it's generally recommended to use String if you need to refer specifically to the class.

e.g.

string greet = String.Format("Hello {0}!", place);

This is the style that Microsoft tends to use in their examples.

It appears that the guidance in this area may have changed, as StyleCop now enforces the use of the C# specific aliases.

Modify your constructor to the following so that it calls the base class constructor properly:

public class MyExceptionClass : Exception
{
    public MyExceptionClass(string message, string extrainfo) : base(message)
    {
        //other stuff here
    }
}

Note that a constructor is not something that you can call anytime within a method. That's the reason you're getting errors in your call in the constructor body.