What is so funny about sin($\alpha+\beta$) versus sin(s+i+n)?
The joke is that there is an ambiguity on whether to read it as the polynomial $s\cdot i\cdot n\cdot(s+i+n)$ or as $\sin(s+i+n)$, as opposed to $\sin(\alpha+\beta)$ which is unambiguous because the arguments of the $\sin$ function are from a different alphabet.
From reading the first answer by Gae. S, it is obvious that some of ambiguity stems from the "invisible multiplication sign". This same author, Edsger W. Dijkstra, has written extensively on the proper choice of notation in other articles, and has criticized the use of this type of implied notation.
There is EWD1059 which I had read previously. Even better is EWD1300 which I had not read yet, but was able to find based on the revelation provided by the earlier answer. This article actually includes another example of this. The author notes that the "invisible multiplication sign" (the dot, when written out) has been hailed as progress through brevity. So, this seems to be another example of him poking fun at the fact that brevity can also yield ambiguity.
(Several weeks after asking the original question, found EWD1115, which is not yet transcribed or searchable on the UT website, but which contains another partial explanation for this on page 3.)
In the context of his other writing, Edsger W. Dijkstra would say that a complicated problem can be made simple by an appropriate choice of notation (and vice versa: poor notation can make a simple problem unwieldy or practically unsolvable). While his examples are seldom laugh-out-loud "funny" and may not quality as comedy, some of the humor is derived from showcasing mini-disasters that could have been easily avoided.
An additional explanation came to me unexpectedly later last evening, when I picked up the book "Surely You're Joking, Mr. Feynman!" for the first time. At the end of the very first chapter he writes:
While I was doing all this trigonometry, I didn't like the symbols for sine, cosine, tangent, and so on. To me, "sin f" looked like s times i times n times f!
Simply stunning.