There's been a lot of discussion about the difference between rationality and post-rationality. Gordon Worley sees the difference being that post-rationality is based upon epistemic circularity, but Eliezer proposed that too. David Chapman defines it in terms of being meta-systematic instead of systematic. Dross Bucket seems to focus on the limits of formal systems. Agile Caveman characterises post-rationality as being aware of the social dynamics within rationalism and which created rationalism, in particular how rationalism was born out of a conflict with non-rationalist and how their on-going efforts to optimise the world entangles them in conflicts (I haven't seen this take anywhere else). And Orthonormal described post-rationalists as those who believe that the rationalist attempt to systematise all knowledge is misguided. Further post-rationality is often painted as being in opposition to rationality, to the extent that I've even heard one post-rationalist describe rationality as a cult.
The most natural way for me to define rationality and post-rationality is as two clouds in the space of ideas and practice. Since Less Wrong style rationality includes some post-rational elements (nameless virtue, system 1 and system 2, ect.), we will define a perspective which we'll call Archetypal Rationality in order to provide a contrast to Post-Rationality. By comparing to Archetypal Rationality instead of Rationality, we'll avoid the temptation to misrepresent and oversimplify Rationality to make the contrast more obvious.
Archetypal rationality is not rationalism in philosophy. It focuses upon the power of the human mind, not unassisted, but in combination with empirical evidence. It has faith that rationality always wins. Proponents likely identify as skeptics, see little value in continental philosophy (or even any kind of philosophy), and are extremely skeptical of claims like Enlightenment. They are likely to make long lists of fallacies and biases, to emphasise logic over emotions, individual reason over tradition and seeks to provide single unified systems. They are likely to prioritise published research over common sense. Advice is likely to consist of a set of rules; do X in situation Y. Discussions of language are likely to focus on producing precise definitions, but attention will also be paid to any obvious connotations.
In contrast, post-rationality believes that the barbarians are likely to win. Continental philosophy is seen as often containing insights, even if they aren't always clearly and precisely articulated. More broadly, it is more acceptable to gesture at a concept even if you can't define it precisely. The default assumption is that there is some instrumental value in traditions and spiritual practises (steelmanning divination). Labelling an argument a fallacy is often seen to misrepresent a person's social action and the label bias is often seen to neglect ways in which it is functional. Frames and ways of being (see virtue ethics) are seen as more important than rules. When discussing definitions, more attention is paid to psychology and there is a much greater focus on meaning as use, as opposed to dictionary definitions. There is a much greater reluctance to dismiss people as irrational as social dynamics are much better understood. Proponents are likely to approve of non-violent communication, circling, meditation and spiritual traditions.
Traditional rationality leans heavy towards archetypal rationality, with elements of post-rationality. Post-rationalists lean heavily towards post-rationality, but may also have some elements of archetypal rationality as well. Less Wrong has tended toward being more influenced by post-rationality, which also makes drawing a distinction betweeen rationality and post-rationality less clean. Post-rationality is open to more ideas, but has a greater risk of going astray.
There's often a complication in defining post-rationality where someone says "post-rationality involves X" and then the other person says "X is compatible with rationality too" and then the cycle repeats. Overall, it is hoped that this framing prevents the tendencies of these discussions to end up in with people talking past each other in this manner.