It's tellling me I'm having an error on closing I believe.
In Nero the option "Finalize CD" might be the answer. If you are burning a large, disc sized iso, I believe it is a required setting to properly close the disc. I am not familiar with the one that you are using now, and don't have any recommendations for settings or it's use in general, so I can't help you there.

Any way of diagnosing a possible problem with a burner?
You can always send it back to the manufacturer or distributor if it is under warranty. Since you said you have had it for two years, that probably is not the case. If it's not under warranty, it might be more cost effective to purchase a new one. They can be had for as little as $30-$40 occasion, in USB 2 configuration. I recently got a deal on one for about $22, plus shipping. There are a few things that you can do to troubleshoot your drive yourself, but it is time and detail intensive. Since you have an external drive, it could be anything from a bad USB cable to the alignment of the laser or a piece of dust on the lens. A firmware update might also be in order, if one is available. You might try to burn a different disc to less than full capacity and verify that one. You can use the VSO disc inspector independent of the burning process and then try to use the CD rom drive that you are having read errors on, to verify the disc too. That might verify if the reader is possibly OK also.