@jab1, if your friend can copy these disks then you need to ask yourself why she can do it and you can't!
A) Because she has a better drive / firmware than you.
B) She uses different software.
C) Both of the above.
From what you and others have said the cause is likely to be copy protection simply because you can't even play them on your current device. So when you're at your friends ask her to burn the disks using the Ashampoo software you're using so you know it's the device that's the issue and not the software.
If this proves to be the case then you're justified in getting an external replacement.
Also, another option is to see if there is a BIOS update for your own PC first apply that then see if the disks will play.
@Anonymous I know why she can - and I can't. She is using an XP machine (yes, I know it's old/unsupported &c) but it works for her, and the version of WMP on there can be configured to bypass DRM, something not possible in W7 and above.
I will look at the option of an external drive if I get a large number of other discs my current machine doesn't like, but as the problem only appears to be with discs from this label and period (I have problem with any other), once we have ripped these the issue can at least be 'parked'.
As far as the BIOS is concerned, I had already looked at that - no dice - but thanks for the suggestion.