I've been pondering this question for a while now, and I have at least a partial answer thanks in part to Nick Saban. Nick clued me in with his snake oil salesman style, and his buddy Bill Belicik is of the same ilk.
See, here's the thing. There is just so much talent to go around. Yes, some players are smarter, and some are faster. Bill focuses on the ones who are smarter. Maybe they have other attributes, but they also understand the game. And he happens to have a QB who makes very few mistakes.
And coaching for the most part is coaching. Is Bill a better coach? No, but I believe he gives his teams his players two edges: (1) he gets them all to play as a team, and no one gets paid more than anyone else (with a few possible exceptions like Tom Brady). So, there's a certain amount of amicable camraderie that happens because they are a team. He lets them leave if they want to make more money, but with his coaching style they succeed, and often want to hang around. They money is good, but not as good as elsewhere, but they're always competitive.
And (2) Bill understands the rules of the game and uses them and/or exploits them. And because he has smart players, he gets them to to understand the rules and take advantage of them, too. Let's go back to a game a few years ago where the Colts travelled to Boston for a playoff game. In that game, every Colts receiever was mugged, held, or otherwise interfered with. The Colts argued, but the refs "let them play." Bill knew that the officials didn't usually flag interference in the playoffs, so he gameplanned for it. Was he right or wrong? Who knows, but he exploited the rules.
Look at the playoff game they played against the Colts this year. If you watch carefully, on the TD the Pats lineman scored before the half, you'll see Maroney fumble, and a wideout comes in and rather than dive onto the ball, he swats it toward the endzone. Do you think that was coached? I do. If they get away with it, its a TD. If they don't - and the odds say they will get away with it because he was diving on the pile - its a penalty and they get another try. Where's the harm?
And so, I believe he coaches his players to play just to the edge of the rules. Play to the whistle. Everyone be active on every play or you hit the bench. And here are some specific rules you want to try and work on (so if it says the o-lineman can grab a jersey and user leverage in this way, they do just that, and maybe a little bit more). It gives him the advantage.
And it gives his team an advantage when they play everyone else who is *following* the rules. 31 teams the same. One slightly different. Its easy to see why he can literally pluck players off the street and have them succeed.
Its not coaching. Its not really cheating. Its maximizing the potential under a set of rules the way no one else does. The only way to overcome it is for another coach to employ the same tactics, or for the NFL to watch for it and start calling them closer or fining him for certain tactics.