What do you mean specifically? You seem to have a good handle on this, so I'd like to know what parameters you were using and perhaps why. I hate to admit I haven't patterned my home shotguns. Note to self...
Going off the advice of some gentlemen more seasoned and experienced in shotgunnery than me, I like a 25 yd pattern about the size of a sheet of notebook paper.
Remington Managed Recoil 00, Remington Express (standard 9 pellet load) of 00 both do this well out of my guns, but Federal LE 127 00 does this in spades. I haven't tried many #4 loads, but the ones I have patterned, the pattern gets very wide very fast. By 10-15 yds the pattern is well off the paper.
Better target coverage? Especially important if you are only getting, say a shoulder, arm (with gun of course) and part of a head presented? Less chance of overpenetration? Still getting 15" of ball. gello penetration at across the room distance, so good enough but not as much over penetration?
I don't particularly worry about over-penetration. Inside my house I'm not shooting toward my kids bedrooms with anything even if it were birdshot. Anything that can penetrate people can penetrate drywall and then people. It's not an easy problem to overcome by switching ammo. The good news is my house is laid out in such a way that I'll not likely be placed in a situation where the badguy is between me and my kids. I also don't have to worry overmuch about neighbor's houses. If I have to shoot straight down my hallway though, my plasma TV and my truck are likely both toast
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As to better target coverage, If only a shoulder or arm are presented it will be nigh shredded with a fist size pattern of 00 buckshot. The #4's I've tried spread too fast for my liking.
Everyone's situation and needs are different. 00 works for me out of my guns in my house/on my property. I've also killed quite a few deer and coyotes with 00 buck and have seen firsthand the performance on flesh. I've never had the opportunity to do the same with #4's.
Presently the bandoliers and shell belts around here are loaded with Remington 00. The HD shotguns in question are loaded with the Federal Flite control 00 though. The bandoliers have been full of the Remington stuff for awhile and I haven't rotated it through. I have no doubt it will work if called upon.
I've really come to like the Federal flite control shells. I've got about 350 rounds of the standard 127 00 (2 3/4" standard load). Remains of the last case I bought plus one that just arrived last week (ammunitiontogo.com). I've also started buying the 3" loads for deer. Killed one a couple weeks ago with it at 40 yds. Dropped in his tracks with his neck and shoulder full of buckshot