That's an auto property, not an anonymous property. There is, in fact, a private backing field for it, it's just generated automatically by the compiler and isn't available to you at compile time. If you run your class through something like Reflector (or examine it at runtime with reflection), you'll see the backing field.
To answer your question of "What's the difference?", the obvious answer is that one is a field, whereas one is a property. The advantage to using auto properties is that it gives you the flexibility to move to traditional properties later, should the need arise, without changing your API. As far as third party code being able to "reach" one but not the other, that would be a question best answered by the other developer. That being said, most API's are designed to work on properties, not fields (since conventional wisdom is that you do not expose fields outside of the declaring class). If the third-party library is reflectively scanning your class, then it's likely only looking for properties.
The important thing to remember is that:
private string backingField;
public string Data
{
get { return backingField; }
set { backingField = value; }
}
and
public string Data { get; set; }
Are compiled to essentially the same code. The only substantive difference is the name of the backing field.