A key takeaway for me is that computers make it possible to have unambiguous notation. Quoting from Iverson's paper: "The thesis of the present paper is that the advantages of executability and universality found in programming languages can be effectively combined, in a single coherent language, with the advantages offered by mathematical notation."
Ambiguity is a real problem with much of conventional mathematical notation, which has evolved in fits and starts throughout history. Mathematical symbols are used, reused, and overloaded with different meanings again and again, in so many ways, that context is often necessary for understanding. Ambiguity hampers the use of mathematical notation as a tool for thought.
The other big takeaway from this paper, for me, is that succinctness makes it easier to reason. That is, programming languages that enable us to express more with less code make it easier for us to reason about -- and with -- code.