"But to people not already familiar with what’s going on the same patterns are not very illuminating; they are often even misleading. The language is not alive except to those who use it." struck me as significant, and "The standard of correctness and completeness necessary to get a computer program to work at all is a couple of orders of magnitude higher than the mathematical community’s standard of valid proofs." is the closest I've ever seen to someone stating the one of the most important differences there.
I would like to see more accessible exposition of modern mathematics to the interested public. There seem to be two opposing forces - assuming more prerequisites lets you talk more easily and go into more depth, but cuts your target market very quickly. Is this an inevitable dilemma?
The problem is the devil is invariably in the details. It is almost impossible to say something useful about category theory starting from even a modern undergraduate mathematics education in less than about a week, because you need to talk about definitions and utility lemmas before that (and category theory is so abstract it's hard to develop an intuition for it). A lot of stuff like topology is more approachable but it's a lot like popularizing the climate research, although less politically loaded; the real theory is much more nuanced than any simple description could be.
If you are interested in this topic and you haven't already done so, check out "The Mathematical Experience"