Hmm. What version of Ubuntu are you using? I use SmartMonTools almost daily to check hard drives and have recently been doing new Ubuntu 12.04 LTS installs and have had no issues installing it. I use the GUI interface called GSmartControl into SmartMonTools. Close Ubuntu Software Center, synpatic if you have it installed, open a terminal, and type:
Of course, if you are using an unsupported, old version of Ubuntu, the repositories may no longer be accessible and you may not be able to install. You could always download the source and build it yourself (I have done that).
sudo apt-get update
sudo apt-get install gsmartcontrol
And with an external hard drive you'll most likely need to issue extra options like
to smartctl to get it to find the USB drive.
Still, SmartMonTools only checks the S.M.A.R.T. health of a hard drive - very useful, but that does nothing to check the file structure, which is what you probably are most worried about here. If the partition is ext3/4 you can always un-mount it and run
(replace "sdb1" with whatever your partition really is on the external drive) You can force a check of the filesystem if it says "clean" but beware that this could take a while to run.
sudo e2fsck /dev/sdb1
Finally - try rsync in "dry-run" mode to compare the original that you copied and the new:
rsync -av --dry-run /original/directory /media/EXTERNAL/directory
(replace "EXTERNAL" with whatever the mounted name of your external drive is)
This doesn't copy anything but shows you what would happen if you wanted to (get rid of the --dry-run switch to copy for real). If you copied everything correctly, you shouldn't have any files listed by rsync at this point.
Depending on how you copied the files, the file creation times may not match between original and the copy. You can issue the option --size-only to compare without checking times.