Hi idefix,
Actually we have. We've heard everything from the wrong bit version of Thunderbird (ie. 32-bit vs 64-bit) to failing to install add-ons being the culprit. In our experience, it's best to proceed as follows when migrating a Thunderbird profile to another PC or environment:
1) Install Thunderbird from scratch on the new PC
2) Install the add-ons you had running on the old PC
3) Run "thunderbird.exe -ProfileManager" in a command prompt to load up Thunderbird's profile manager
4) Create a new profile (that will become your migrated profile) and point the location of the profile folder to where you'd like to store it going forward (this step makes important settings changes to your profiles.ini file).
5) Copy your old Thunderbird profile folder over this newly created profile folder, excluding the extensions directory.
Load up Thunderbird with the new profile and you should see your panels restored. In most cases, it's the extensions directory that causes problems.
If that doesn't work, you can try copying over only your "Mail" folders - then at least you'll have your data (backed up emails) loaded.