R – oracle without installing the data provider

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.

There is actually a (subtle) difference between the two. Imagine you have the following code in File1.cs:

// File1.cs
using System;
namespace Outer.Inner
{
    class Foo
    {
        static void Bar()
        {
            double d = Math.PI;
        }
    }
}

Now imagine that someone adds another file (File2.cs) to the project that looks like this:

// File2.cs
namespace Outer
{
    class Math
    {
    }
}

The compiler searches Outer before looking at those using directives outside the namespace, so it finds Outer.Math instead of System.Math. Unfortunately (or perhaps fortunately?), Outer.Math has no PI member, so File1 is now broken.

This changes if you put the using inside your namespace declaration, as follows:

// File1b.cs
namespace Outer.Inner
{
    using System;
    class Foo
    {
        static void Bar()
        {
            double d = Math.PI;
        }
    }
}

Now the compiler searches System before searching Outer, finds System.Math, and all is well.

Some would argue that Math might be a bad name for a user-defined class, since there's already one in System; the point here is just that there is a difference, and it affects the maintainability of your code.

It's also interesting to note what happens if Foo is in namespace Outer, rather than Outer.Inner. In that case, adding Outer.Math in File2 breaks File1 regardless of where the using goes. This implies that the compiler searches the innermost enclosing namespace before it looks at any using directive.