The title says it all. Sometimes it seems that the Name and x:Name attributes are interchangeable.
So, what are the definitive differences
When you declare a Button element in XAML you are referring to a class defined in windows run time called Button.
Button has many attribute such as background, text, margin, ..... and an attribute called Name.
Now when you declare a Button in XAML is like creating an anonymous object that happened to have an attribute called Name.
In general you can not refer to an anonymous object, but in WPF framework XAML processor enables you to refer to that object by whatever value you have given to Name attribute.
So far so good.
Another way to create an object is create a named object instead of anonymous object. In this case XAML namespace has an attribute for an object called Name (and since it is in XAML name space thus have X:) that you may set so you can identify your object and refer to it.
Conclusion:
Name is an attribute of a specific object, but X:Name is one attribute of that object (there is a class that defines a general object).