DVD and CD Drives have small lenses though which the laser beam is focused on to the disc itself.
I imagine that a composite DVD/CD drive will need separate lenses for each type of disc.
If a lens get blocked or dirty then it can't see the disc so it won't be able to either play it or write to it.
Before you involve an expensive computer shop, either buy or borrow a lens cleaning disc. This is like an ordinary DVD disc but has small brushes on it to wipe the lenses clean.
I got one from the Pound Shop a couple of years ago so they are not expensive and can be very useful.