Woohoo! Found the solution when tracking down that error!
The DCOM configuration was not correct for some reason. Don't know how it got changed, maybe it's something the developer did when they installed the application. Anyway, this is the procedure I ran to fix it:
1. Click Start, click Run, type Dcomcnfg, and then click OK.
2. Expand Component Services, and then expand Computers.
3. Right-click My Computer, and then click Properties.
4. Click the COM Security tab.
5. Under Access Permissions, click Edit Default.
6. Verify that the following accounts are listed:
Comparing the broken server to one that worked, the broken server had Everyone listed with local and remote access. The one that works did not have Everyone listed. I removed it in the broken server and hit apply. Everything else looked OK.
Step 2: Verify DCOM default properties
1. Click the Default Properties tab.
2. Verify that the following configuration:
• The Enable Distributed COM on this computer check box is selected.
• In the Default Authentication level box, Connect is selected.
• In the Default Impersonation level box, Identify is selected.
3. Make any required changes, and then click OK.
4. Restart the computer.
In this case, the broken server had Default Impersonation set to Anonymous. I changed it to Identify and it gave me an error when I clicked OK saying I didn't have the permissions to do that. However, the change took, I didn't have to reboot at all, and now MMC opens fine and Windows Update works fine too. This problem is RESOLVED! Thanks again!