Are using iCloud by any chance? If you are, importing .abbu files does not work. If you watch closely you will see the contacts do get imported, then immediately deleted by iCloud.
Actually, the same tip is probably correct, if I read correctly. The new Contacts.app in Mountain Lion probably has made some changes to the .abbu format, which means when the file is opened on the old Mac, it doesn't read the data properly. Exporting from the new machine as a VCard file should work properly, although you'll have to do groups individually (or recreate them afterwards).Are using iCloud by any chance? If you are, importing .abbu files does not work. If you watch closely you will see the contacts do get imported, then immediately deleted by iCloud.
Go back to the old machine and do a command-a to select all the messages, then in the export menu select the top option to export as a .VCF file. That will import into Contacts with iCloud on without issue.
If you are not using iCloud, ignore everything I just said.
If I'm reading the OP correctly, though, they're going the opposite direction. Of course a Lion .abbu file would be useable in Mountain Lion, but it may not work in reverse, particularly if it's further back. If they're transferring contacts from their new Mountain Lion computer to a computer running Tiger, I can definitely see it not working, but a VCard file should work fine.
