The difference I would point out, is that Rust has corporate sponsorship. I don't recall any large corporations sponsoring e.g CMUCL/SBCL to the same level.
In the case of Lisp, it was done in by two things: AI winter, and the fact that the Lisp community was never able to organize itself. This is the famous "Lisp curse": it is precisely the fact that Lisp is a productivity multiplier that seals its fate because it allows individuals to get things done without collaborating.
It was done in / seals its fate
https://www.marktarver.com/bipolar.html
I always thought this essay was a great take on Lisp(ers), but it's interesting how those sort of statements eventually can self-perpetuate a couple negative events into a state of learned helplessness.