Optically, they are identical. However, the laser disc player uses an analog recording technique (non-digital) to recreate the video signal. I think that the lengths of ridges (or perhaps pits) in the aluminum surface are used to control the analog signal strength.
The optical system of the CD player must be very clean. If the final lens has dust on it, the photodiode will not see the full range of light and dark patches that it expects. A cleaning CD presumably cleans this final lens, although I’m not sure how. In principle, the whole CD player should be pretty resistant to dust problems because the laser beams are large except when they focus on the CD itself.
I only know how the prerecorded mini disc players work: they work a lot like CD players. However, they use a much smaller disk, made possible by intelligent data reduction. Instead of using 16 bits to represent each current measurement, the mini disk uses a variable number of bits. The recording equipment determines how many bits are needed to represent the sound accurately and eliminates unnecessary (or inaudible) details in the current measurements. The optical systems in mini disc players are the same as CD players.
Most of the CD’s you encounter are prerecorded. These CD’s were mass-produced from a master, using plastic molding techniques, followed by metal deposition and painting. Recordable CD’s, which are used now in CD-ROM applications, are written by an intense laser beam, which alters the reflectivity of the CD spot by spot to create a disk that behaves just like a prerecorded CD. However, once a CD has been “written”, it cannot be cleaned for rewriting. At present, recordable CD’s can only be written once. There are some new optical and magneto-optical techniques around that allow erasure, but I don’t think these techniques have appeared in CD’s yet.