One very important difference that hasn't been mentioned yet is when you are shooting animals such as insects or other arthropods. They are light sensitive. If you want to have narrow aperture (for getting good depth of field) and high shutter speed (for freezing movement), you need a whole lot of light. And most animals will not stay in place given a whole lot of permanent light: you want it pulsed, namely triggered by the camera. Some LED lights can do this as well, but usually needing quite longer exposure times for comparable results. As a side remark: for things like insect photography, there may be a point in not using TTL for a digital camera but rather the less precise "autothyristor" mode of a flash where the flash itself does the metering: digital camera TTL tends to rely on a pre-flash for the metering that may cause animals like jumping spiders to move before the main flash engages.

Another difference is the light spectrum. Many flowers have some very intense pigments and those tend to fall into narrow lines of the color spectrum. A flash has a continuous light spectrum and thus creates a comparatively natural color response while the result with LED lights may be less convincing.