Part of Penrose's conviction that his reasoning is intrinsically more powerful than that of a computer program may come from his using kinds of reasoning that he implicitly denies machines. There are two such kinds of reasoning.
The first is that he reasons about theories in general, i.e. he uses variables ranging over theories. As far as I can see he never allows for the computer program doing that. However, reasoning about theories as objects is not different in principle from reasoning about other objects.
The second is that much of Penrose's reasoning is nonmonotonic, e.g. preferring the simplest explanation of some phenomenon, but his methodology doesn't allow for nonmonotonic reasoning by the program. Mathematicians' acceptance of the axiom of choice, for example, occurs through informal nonmonotonic reasoning. Formalized nonmonotonic reasoning is a recent development.