The dye doesn't really affect quality much.
There are a few differences between them: pthalocyanine (very pale goldy-green) is the longest lasting - it is supposed to be able to last about 100 years, and is the only dye acceptable for use in archive quality CDRs. This dye is also better for use at very high recording speeds.
Cyanine (moderately green, blue, or turquoise - depending on formulation) is more common, because it is the original CDR dye, and is probably slightly cheaper. Its archival qualities are not as good, although some disc manufacturers quote a useable lifetime fo 75 years. This dye performs less well at high speeds, although newer formulations have improved this.
Azo (very deep blue) is quite rare - only manufacturerd by Mitsubishi, it is used in few discs (notably Verbatim). Its properties are similar to cyanine (hence the common mistake, that the two dyes are the same), and it is treated as such by CD-R drives. However, it is more stable than cyanine, has better UV resistance, and a longer life.
For most uses, there is little point in chosing one dye type over another. The brand is the most important; the best manufacturers go to lengths to ensure that the discs meet the appropriate specifications, and that the top coating won't peel off.
i once bought some cheap CDRs, and the quality was terrible - the markings on the top side had been burned into the dye (too much UV light during the lacquer curing phase), the top side peeled off in your hands, and there were air bubbles in the plastic on the recording surface!