Always finalize a disc. A CDR and CDRW only differ in that the CDR is one write only, meaning you can't reuse it. You can replace the information written to a CDRW. Finalizing a CD is what writes a T.O.C or Table of Contents to the disc. This is what tells a CD Player: how many tracks, where they are divided, and any other pertinent information (CD Text, etc. etc.). If the disc isn't finalized, the deck doesn't know how to play the CD or what's on it.
That being said, maybe your deck isn't compatible with an MP3 incoded information. Have you tried a few different brands of media? As different companies have different opinions on the respective reflection properties, dyes differ from company to company and certain dyes work better with certain cd players. You can identify the difference between various dyes by the color on the underside of the disc. You'll just have to experiment.
If your deck plays a CDR with WAV. info consistently regardless of brand of CDR, but have no luck with MP3 info, you just have a defective technical data sheet.