The one thing that's different is on the working Mac, it also shows dots for an entered password. If I copy my server password from 1Password and paste it into the password field in Hazel on the non-working Mac and try to connect, it says:
The folder “transcripts” could not be accessed. The operation couldn’t be completed. (NSURLErrorDomain error -1012.)
Paul studied my log files (quite a while back) and wasn't able to figure out what's wrong. I figured it was worth posting here to see if anyone else had an idea.