The term 'CDI' is often, incorrectly, used to describe electronic ignition systems. Most modern ignition systems are actually Inductive Ignition systems for good reason, especially when using lean burn fuel mixtures. Inductive ignition systems can provide prolonged spark duration; resulting in more reliable and a cleaner burn in modern lean burn engines (see diagram 3).
Capacitive discharge systems may have advantage in older 4-stroke engines, an engine running beyond its service life, or a cheap 2-stroke engine. These engines will be running an oil rich / fuel rich mixture, which may cause fouling of the spark plug gap. The higher initial discharge of a CDI system may be able to 'burn' off these deposits better than a comparative inductive ignition system.