Different drives certainly have different reading capabilities. This feature is most evident with low quality discs, like Sony branded media: when a drive is rather picky, it is not able to read well bad burned discs.
You can try to improve reading capabilities updating the firmware, but even if this often improves the drive performance there is no guarantee that you'll be able to read these discs.
The Liteon 160P6S is able to run the quality test with CD-DVD speed. Can you try to run the test and post the picture here?
Can you try anyway to do a test with a Verbatim disc? If also Verbatim fails, then the drive could be effectively damaged.
To completely exclude some sort of software issues in your computer, can you test the drive in a different computer (maybe you can ask a friend to run a quick test)?