Are you using different brands of -R between the two drives?
Are you burning at a different speed between the two drives?
Are you using different software to burn / rip / compress?
Most old standalone players only like -R discs or none at all and it just could be that the older drive burns at a slower speed and hence that is the problem.
Have you tried verifing the disc to see if it has many errors on it as it could be the new drive doesn't like the media your using.