The linked paper from 1993 contains a serious critic of practices in theoretical physics at the time:
This unreliability is certainly a problem in theoretical physics, where the primary literature often becomes so irrelevant that it is abandoned wholesale. I. M. Singer has compared the physics literature to a blackboard that must be periodically erased. Physicists traditionally obtain much less benefit from the historical background of a problem, and they are less apt to search the literature. The citation half-life of physics papers is much shorter than in mathematics.
A similar critic is applied to the practice of publishing research announcements:
Some areas in the Russian school of mathematics have extensive traditions of theoretical work, usually conducted through premature research announcements. From the numerous possible examples we mention only two. [...] 1954 Kolmogorov announced [...] proofs were achieved by Arnold in 1959 for the analytic case and by Moser in 1962 for the smooth case.
[...] In 1973 the respected mathematicians Dobrushin and Minios published an announcement of that result. Two years later when no indication had come from the Russians of a proof, Glimm, Jaffe, and Spencer resumed their work on the problem and eventually gave two different proofs. A couple of years after that Dobrushin and Minios published a retraction of their original announcement.
The paper points out the devastating consequences similar practices in mathematics had in the past, and tries to propose solutions how to avoid those consequences and allow theoretical work:
Theoretical work should be explicitly acknowledged as theoretical and incomplete; in particular, a major share of credit for the final result must be reserved for the rigorous work that validates it.
Only one solution is proposed for research announcements (revealing the authors opinions):
Research announcements should not be published, except as summaries of full versions that have been accepted for publication. Citations of unpublished work should clearly distinguish between announcements and complete preprints.
I was not able to access the 2015 Hirzebruch Lecture in Bonn. It would certainly be interesting whether Jaffe believes that the problematic practices still persist, or whether he was rather reporting on the success of his proposed solutions.
Even if this may not be the answer the questioner has hopped for, the question prominently features that link to the 13 page opinion piece and a link to an unavailable lecture. Hence it should be allowed to point out that the opinion piece contains strong statements which overshadow (even so they may be true) any potential discussions about details of the suggested analogies used in the opinion piece.
Jaffe justifies to say "experimental mathematics" for the activity of finding proof of theorems via:
A relevant observation is that most theoretical physicists are quite respectful of their experimental counterparts. Relations between physics and mathematics would be considerably easier if physicists would recognize mathematicians as "intellectual experimentalists" rather than think of them disdainfully as uselessly compulsive theorists. The typical attitude of physicists toward mathematics is illustrated by a passage from a book of P. W. Anderson, "We are talking here about theoretical physics, and therefore of course mathematical rigor is irrelevant and impossible."
This makes it clear that Jaffe is talking about sociological phenomena here. Because no comparable phenomena exists (or existed) within the mathematical community itself, there was never any need to voice this kind of sentiment by type theorists such as Martin-Löf. The close relationship between verifiability, falsifiability and meaningfulness on the one hand, and experimental observation in physics, rigor and proof in mathematics, and absence of metaphysical speculations in philosophy has been clearly voiced by proponents of logical positivism.