The first answer pretty much covered it all, but I'll add three personal interests:
a) I like the idea that the building was its own scaffold. The structural and constructional logic meant that it could largely support itself as it was constructed. For example the spiral stairs were built as a series of stone steps inside a stone drum that required no temporary or additional support.
b) Gothic architecture, particularly its churches, was seen as 'organic' because few are pure or perfectly unified in their design and expression. However, there was a great deal of design involved and they are highly rationalized structures. The fact that there was a lot of trial and error involved doesn't mean they were intuitively designed.
c) Finally, what I most appreciate about the cathedrals, in particular, is that although they contain a great deal of visual meaning - stories told in stain glass, sculpture, and through metaphor - they incorporate spatial meaning through bodily rituals. That is, they make the space of the architecture important. The procession of moving of moving from the profane to the sacred is embedded in the linear plan and you experience it by walking. The effect of parallax, accentuated by the many columns, heightens the sense of transition. My favourite spatial detail is the central column (often called 'Christ the column') found in many of the central portal doors that physically makes one step off the central axis (which was considered sacred) - the act of deferring is not something that is read, graphic or visual, but a bodily experience, embedded in the organisation of the architecture itself. This kind of spatial significance is lost afterwards and doesn't begin to appear again until the 19th century.