Problems with Catalog

The new laptop is looking for a Tally .tly file that I don't see in the backup files.

Generally that means that the backup did not run completely. The 'backup.tly' file is the last one to be written to the disk.

I tried to do a full backup on the old laptop and a restore on the new laptop, but the files don't seem to match. 

All the files saved in the backup folder are renamed copies. The backup folder can't be used as a copy, it has to be 'restored',.

Would it be possible to do a new backup, waiting some time after the last message, to be sure everything is written?

The restore won't work if you don't have the backup.tly file in the backup folder.