The answer: "There isn't one."
It seems a time honored tradition that you start a young hunter out with the diminutive .410 shotgun. In days past it was typically a single but pumps and even a couple of autoloaders have been manufactured. The rationale behind the choice is that the light recoil will encourage better shooting. If it were a rifle, I'd agree. But it's a shotgun, so I don't. Not at all.
Here goes the long winded explanation. The shotgun kills with pellets. The number of pellets gives you pattern density. Pattern density dictates how far away you can kill stuff. Simple? Not really. The typical .410 load of #7.5 shot is 1/2 to 11/16 ounces. The typical 20 gauge is firing 7/8 to 1 and 1/4 ounces of shot. That's 175-241(.410) compared to 306-437 (20 gauge) individual pellets in an individual shell. Considering it only takes one lethal pellet to kill... the 20 gauge is clearly a more powerful and lethal round. Hands down, no way around it. The 20 gauge will fire more pellets, have higher pattern density and be lethal at a longer range and be way more forgiving.
What about recoil? The .410 firing a 3" load in a 5.5 gun generates 10 ft/lbs of recoil force compared to 16 ft lbs for a light field load in a 20 ga. Is that significant? Maybe if you have your kid shooting 12 rounds of trap every day but for the average kid getting to blast a box periodically and maybe go shoot a grouse or some rabbits... it amounts to nothing. Any kid old enough to carry a lethal gun can stand up to 16 ft/lbs of recoil force, which is roughly equivalent to the 7-08 in a light rifle. There are a number of light loads available and a slightly heavier 20 gauge is going to have less felt recoil than a light .410 gauge.
My first shotgun? A .410. My kid's first shotgun? A .410. The wrong choice on both counts. A generational bad decision that keeps propagating.
My early attempts on squirrels with an ancient, hand me down .410 were abysmal and I quickly learned I was far deadlier with a .22 to much further than the .410 could be counted on. My son's results were much the same- after he let fly at a rabbit at a mere 20 yards and the rabbit ran off he was dejected. Dirt flew up all around the rabbit but he never left a drop of blood. He was simply in a hole in the pattern of the feeble payload. A 20 gauge would have provided one dead rabbit. It was shortly after this affair that I outfitted my son with a small frame 20 gauge pump and his lethality in the field went from zero to 100% in a single season. He took squirrels, hares, grouse, ptarmigan and loved that if he could get within thirty yards- he could seal the deal. He never made a comment about the recoil either. I've long held that recoil is only felt at the range. On game, I can't recall anyone ever talking about recoil- even at stout levels.
The .410 is very pleasant to fire, but lacks the killing power to give kids a good chance on game. I feel you're better off letting them feel some thump and give them some field success than playing softball on recoil and giving them a lot of frustration in the field. I think as a community we get this wrong all the time.
So what good is the .410? The .410 is a wonderful little cartridge and can make a wonderful light shotgun in the hands of a pro that points like a magic wand. In the hands of a kid it's a frustration, in the hands of an old master- it's a joy. The light gun weight and low recoil make shooting pleasant and the lack of pellets provide the level of challenge that many accomplished folks crave. Kids are still learning the basics and nothing like field success encourages more practice.
The .410 is widely promoted as ideal for beginners and that's all wrong in my book. We should be promoting it to the masters, right alongside the 28 gauge.
For a kid- the 20 gauge firing light loads is where it's at.