E-TTL II is usually the better general-purpose mode because it meters through the lens and can use lens distance information. In practice, that makes it more accurate in situations where the flash sensor’s view doesn’t match the camera’s view—such as close-ups, small subjects against a background, or off-camera flash where the flash isn’t pointed from the same position as the lens.

Ext.A uses the flash’s own sensor, so it reacts very quickly and consistently and doesn’t need a preflash. That can be useful because E-TTL II’s preflash can sometimes trigger blinks, and the extra metering step may slightly slow rapid multi-shot shooting.

The downside of Ext.A is that it can be fooled when the flash sensor is “seeing” something different from what you actually want exposed. It may overexpose a small subject if it is trying to also light the background, and it becomes less reliable when the flash is off-camera or not aimed from near the lens axis. It can also be problematic with non-coupled/stopped-down lenses unless aperture information is entered correctly.

So practically: use E-TTL II for most situations; use Ext.A when you want simple, fast, no-preflash auto flash behavior and the flash sensor has a clear, appropriate view of the subject.