I was going to open a thread about this just now 
Looking at the spread patterns from left to right, first is aimed down sights shot )pretty tight), second is normal shot (about as big as the reticle), third is special ammo (about 3x as wide as the reticle). Special ammo makes no difference regarding ADS as far as I can tell, and it does about the same damage (in total) as the normal shot. It does have a stronger push, though. While not a lawnmower, it’s very useful for thinning out hordes.
What I would like to see is different shotgun variants with different special ammo, like a solid shot for high single target damage even at range, incendiary ammo to set enemies on fire, or melta ammo with very high damage at point blank distances but useless at range.