Pardon me, they aren't synonymous and I am not equating them, but instead relating them in exactly the way that you say: huge swats of experimental mathematics are literally completely part of the basic conventional methodology within dynamical systems theory (analogous to calculus being completely part of the basic conventional methodology in physics). In fact, much of experimental mathematics methodology used in dynamical systems theory tends to be seen as so pedestrian to dynamical systems researchers that they themselves don't even bother to give these methods a name, much less refer to them as 'experimental mathematics'.Sorry, but this is not correct. I am a mathematician working in dynamical systems theory, classified here (2010) and here (2020) by the AMS. Suggesting that "experimental mathematics" and "dynamical systems" are synonyms is wrong.
"Experimental mathematics" can suggest directions of research in dynamical systems theory (and so can many other fields in mathematics and the sciences), and dynamical systems theory can suggest new mathematical experiments (analogous remarks apply), but you cannot equate them.
It would of course be more correct to regard experimental mathematics as a subfield within mathematics itself, but that completely misses the main point I am trying to make, namely that large parts of experimental mathematics (e.g. doing computational experiments, applied bifurcation theory, stability analysis etc) are in fact conventional methodologies within dynamical systems research in actual practice, whether or not dynamical systems researchers use or recognize the terminology 'experimental mathematics'.
In other words, if a physicist wants to know how to use experimental mathematics in his own research it is productive (or at least, it was in my case) to talk to a dynamical systems theorist or to peruse the dynamical systems theory literature, instead of talking to a (pure) mathematician or perusing the mathematics literature which in my experience is quite counterproductive because many mathematicians don't even seem to recognize dynamical systems theory as proper mathematics.