One difference is that the old-fashioned form gives undefined behaviour if the class (or a friend) tries to destroy itself. Within class members and friends, the destructor is accessible, so there is no compile-time error if it's used. Instead, you get a violation of the One Definition Rule. In practice, this will cause a link error, but formally the behaviour is undefined.
Deleting the constructor (or other function) causes a compile error if the function is needed in any circumstances; and states the intent more explicitly, allowing the compiler to give better diagnostic messages. This is especially useful when functions are implicitly deleted due to something else being deleted.