All of the properties of the accounts were the same. Both had entries in the registry that were identical. I looked at the files that they opened using the SysInternals procmon program. I looked at the network traffic that the launcher program generated using Microsoft Network Monitor 3.4.
Procmon clued me in that there were some log files being generated. They are found in C:\Program Files\Windows Small Business Server\Logs. (C:\ProgramData\Microsoft\WSSG\Logs\ in SBS2011.)
Here's the deal. Launcher downloads an executable called connectcomputer.exe and runs it. Connectcomputer does a huge amount of stuff. I'm not sure it needs to do all that--it has the distinct feel of having been written by an intern. One of the things it does is enumerates your user profile directory and checks every file. If any file in the entire directory tree is not accessible by it, or it runs into any problems with anything, it won't list that account.
I opened connectcomputer.log in notepad and searched for the account that was missing. It showed an exception. For some reason, there was a recursive reparse point to a directory in my c:\users\myacct\appdata\local\application data directory that looped back to the same directory. A reparsepoint is a fake directory that links to another real directory on the drive. So I (and connectcomputer) could both cd into the "application data" directory forever until the filename became too long and it threw an error. Thus it was finding "c:\users\myacct\appdata\local\application data\application data\application data..." etc. By the way, this will not show up in the Windows Explorer, but does show up in the command line. How this reparse point got created is a mystery...maybe having to do with the account being originally created in Windows XP.
The next trick is removing a reparse point. I had a clue about that, and it just took poking around a little to find it. There is a nifty command line tool that comes with Windows called fsutil. It allows mucking with NTFS from the command line. First you must cd to the the directory that has the bad reparse point. The magic command line that fixed my problem is:
cd \users\myacct\appdata\local
fsutil reparsepoint delete "application data"