They are roughly in competition, though Deno made the decision early on to largely split itself from the ecosystem and existing conventions of Node (and an enormous body of existing JavaScript packages on npm), whereas Bun did not. Bun, AFAIK, aims to be compatible with Node and existing the npm ecosystem.
I met Ryan Dahl once at like a CouchDB meetup in Oakland or something a decade ago. He was obviously a brilliant guy, and humble, and very likable as a result.
I very much admire that he seems to be trying to get things right in light of lessons learned, but history would imply that backwards compatibility tends to trump better technology on average.
It seems both are attempting to make money by providing their own "serverless clouds" and hoping people will use those, instead of AWS/GCP/Cloudflare Workers/Fly/etc.