Technology is a wonderful thing - shame about the incompatabilities!
When originally released, the formats for domestic players were CD-R, (for CDs!) and DVD-R (for DVDs, obviously!). These were specified and used for pre-recorded media such as the general public would buy or hire.
Only later did the other formats (CD+R, CD-RW, and later the same for DVD) get introduced for differet basic usage - mainly on computers. There are other formats as well, but to get back to the point, you can't play your CD/DVD because it's in a format your player doesn't understand. Although, more recently, multi-format players (e.g. your car player) have become available.
Because your player expects a pre-recorded CD-R, you need to 'finalise' it when you burn it so it becomes 'pre-recorded' and unchangeable. You can't finalise a CD-RW as it's not part of the format definition, so Nero doesn't offer it. (Same applies to DVDs)
So, what to do? Read each of your -RW (or whatever) disks into your computer and reburn them and finalise as -R versions. A pain, I know, but the only solution, unless you buy a more modern player (and check the specs first!)
Just to expand in DVDs and these TV to DVD recorders: As ststed above, pre-recorded (common commercial films etc.) are in DVD-R format. These TV recorders may only record in +R or RW as they are more amenable formats for ad-hoc recording, but the result won't play on common home players. Yes, I got caught by it. If you then edit the - say RW disk on a computer, it may not then play on the recoder .... Hmmm. Betamax, anyone?
My biggest gripe with these recorders is that they aren't point and shoot like the VCR. Why can't they include a battery-backed or static RAM to remember -say- the current position on the last -say- 100 disks so it works the same way: put a disk back in and pick up where you left off??? (btw Flash RAM has a relatively