They look the same. They work the same. But they are not the same. A21 bulbs see more use in commercial settings, whereas the A19 has been traditionally used in more residential applications. This is due mainly to their difference in size, as the A21 is slightly larger in diameter and puts out a bit more light, as well (give or take a few Lumens).
The number that follows the “A” in these bulbs indicates the bulb’s size when measured in 1/8” increments. For instance, a standard A19 bulb diameter measures at 19/8”, or when converted is 2.375”. The A21 bulb measures 21/8”, or 2.625” in diameter, and is also slightly taller than the A19 which may prevent it from properly fitting in some fixtures. Be mindful of this when selecting a replacement bulb for your application.