Reason
Greek nous
Latin intellectus
Reason is flexible. It resists fixed formulation. It is shaped by experience. It involves the whole living being, combining the mental and physical . . . the spiritual and material. It is the intellectual glue of sacramental existence. It is characterized by intuition.
Reason is "congenial to the operations of the right hemisphere." McGilchrist, The Master and His Emissary, p. 330.
Rationality
Greek logos/dianoia
Latin ratio
Rationality is more rigid than reason. It feeds knowledge to specific groups. It's mechanical and governed by explicit laws. It focuses on consistency and proof. It looks for "either/or" answers (not "both/and"). It is the intellectual glue of the expert.
Rationality is congenial to the operations of the left hemisphere. Id.
Reason is Prior
Rationality stands on ground that is not the product of rationality.
Rationality can operate only when the structure is in place, but the structure itself is not formed by rationality. Something comes before and builds the structure, and that "something" comes from processes that are largely intuitive and formed by reason, hence reason is prior to rationality.
"All rationality can do is to provide internal consistency once the system is up and running." Id.
The Cartesian Method is Insane
Descartes starts with the absurd premise that rationality is prior and, therefore, rationality is king. It's why rationality is king of the modern world. It's why the modern world is mad.
Passages from the Beginning of Chapter 14, "Reason's claims on truth," of McGilchrist's The Matter with Things
Reason [Rationality] suggests a linear way of thinking, seeking chains of causation, which makes sense only in a limited environment. Its mode of operation is local, one bit at a time. Reason suggests a global, holistic understanding, which makes sense only in the round. It is a seamless apprehension of the world. Yes, you read that correctly. Reason means different things – has in particular two distinct meanings. In our world, one is becoming predatory, the other disappearing like a hunted animal. . . .
The distinction I am making between two kinds of reason is disguised in English by its using the same word for two very different phenomena. In other languages, it is not necessarily so. In German there is a distinction between Verstand and Vernunft, in Latin between ratio and intellectus, in Greek between dianoia and nous: in the first of each case, what one might call, in the absence of anything better, rationality, a linear process; in the second, reason, an understanding in the round. . . .
Much philosophical reasoning is of the first kind – for two thousand years in the West, between Plato and Kant, it was more or less exclusively so. Elsewhere in the world, and at other times in the West, this kind of thinking, the kind associated more with the left hemisphere than the right, has been considered of only limited use, and a potential source of misunderstanding when given too much credence. According to this point of view, not to temper it with other modes of understanding would itself be contrary to reason. The problem is, however, that once one is inside the rationalist bubble, one can no longer see any need to break out of it, let alone any way of doing so. . . .
Rationality is exclusive: reason is inclusive, balancing rationality with intuition, emotion and imagination. . . .
Reasoning in the sense of rationality is a consistency tool, and nothing more.